on artin’s conjecture for primitive roots · 2002. 6. 26. · resume ´ nous consid´erons ......
TRANSCRIPT
On Artin’s Conjecture for Primitive Roots
by
Francesco Pappalardi ∗
Department of Mathematics and Statistics
A thesis submitted in partial fulfillment
of the requirements of the degree of
Doctor of Philosophy at McGill University
February 1993
∗ c©Francesco Pappalardi - (1993)
ACKNOWLEDGMENTS
There are many people who I would like to thank at this point:
My Master thesis Supervisor Marco Fontana for having trusted me and supported
my choice to come to Canada. Professor Paulo Ribenboim for having always advised
me on my choices and his precious mathematical teaching.
Professors Jal Choksi, Ian Connell, Hershy Kisilevsky, John Labute, V. Seshadri
and Georg Schmidt, whose help and politeness made my time at McGill pleasant.
Raffaella Bruno who corrected the english of this thesis. None of my problems
were ever unsolvable for her.
Valerie McConnell, Elaine Tremblay for never having denied a smile and their
valuable help.
My friends Mark Fels and Djordje Cubric, the first having shown me the human
side of being a graduate student, and the second , with whom I shared an office was
a true gentleman and a solid office-mate. It will not be easy to find others like them.
Masato Kuwata, who has helped me considerably with the computers and Damien
Roy who helped me with the theory of Kummer’s Extensions.
My wife Claudia and my daughter Silvia, the first for support and encourage-
ment during these difficult years, and the second for having waited that extra week
necessary to finish this thesis.
Last but certainly not least I would like to thank my supervisor Ram Murty. He
is an exceptional man, who placed his trust in me and guided me to many of the
problems which lead to this thesis. His teaching did not cover only Mathematics but
he has also been a precious counsellor on any issue that I have brought to him. I
hope to be able to pass on what I have learned from Professor Murty to others in the
future.
ABSTRACT
Various generalizations of the Artin’s Conjecture for primitive roots are consid-
ered. It is proven that for at least half of the primes p, the first log p primes generate
a primitive root. A uniform version of the Chebotarev Density Theorem for the field
Q(ζl, 21/l) valid for the range l < log x is proven. A uniform asymptotic formula for
the number of primes up to x for which there exists a primitive root less than s is
established. Lower bounds for the exponent of the class group of imaginary quadratic
fields valid for density one sets of discriminants are determined.
RESUME
Nous considerons differentes generalisations de la conjecture d’Artin pour les
racines primitives. Nous demontrons que pour au moins la moitie des nombres pre-
miers p, les premiers log p nombres premiers engendrent une racine primitive. Nous
demontrons une version uniforme du Theoreme de Densite de Chebotarev pour le
corps Q(ζl, 21/l) pour l’intervalle l < log x. On etablit une formule asymptotique
uniforme pour les nombres de premiers plus petits que x tels qu’ il existe une racine
primitive plus petite que s. Nous determinons des minorants pour l’exposant du
groupe de classe des corps quadratiques imaginaires valides pour ensembles de dis-
criminants de densite 1.
Contents
INTRODUCTION 3
1 ON HOOLEY’S THEOREM 9
1.1 A generalization of Hooley’s Theorem . . . . . . . . . . . . . . . . . . 9
1.2 Computation of the Densities . . . . . . . . . . . . . . . . . . . . . . 16
1.3 The Main Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 ON THE ARTIN L-FUNCTIONS OF Q(ζl, 21/l) 24
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Artin L-functions of L/Q . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 On the non-Abelian L-function of Q(ζl, 21/l) . . . . . . . . . . . . . . 31
2.4 An Application to Chebotarev Density Theorem . . . . . . . . . . . . 40
3 ON THE NUMBER OF PRIMES GENERATING F∗p 43
3.1 Extending Hooley’s Method . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Relaxation of the Hypothesis and Improvements . . . . . . . . . . . . 48
3.3 A Density One Result . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 MORE ON PRIMITIVE ROOTS 60
4.1 Another Generalization of Hooley’s Theorem . . . . . . . . . . . . . . 61
4.2 Calculation of the Densities . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 An Application to the Least Prime Primitive Root . . . . . . . . . . . 80
1
APPENDIX A: ON DIVISORS OF p− 1 83
APPENDIX B: ON THE EXPONENT OF THE IDEAL CLASS GROUP
OF Q(−d) 90
APPENDIX C: OPEN QUESTIONS AND FUTURE RESEARCH 99
Variants of the Bombieri-Vinogradov Theorem . . . . . . . . . . . . . . . . 99
The Lang-Trotter Conjecture for Abelian Varieties . . . . . . . . . . . . . 101
REFERENCES 104
2
INTRODUCTION
The famous Artin Conjecture for primitive roots states that any integer a 6= ±1 which
is not a perfect square is a primitive root for infinitely many primes. More precisely,
if Na(x) is the set of such primes up to x, then
Na(x) ∼ A(a)π(x)
where A(a) 6= 0.
Artin also gave an explicit formula for A(a) and his intuition was based on the
following heuristic argument (see [1]):
For any prime p less than x, let Pa(q) be the probability that the prime q divides the
index [F∗p : 〈a〉]; then,by considering such instances independent, we have
A(a) =∏q
(1− Pa(q)).
In order to have q|[F∗p : 〈a〉], the two necessary and sufficient conditions
p ≡ 1(modq) and a(p−1)/q ≡ 1(modp) (1)
must be satisfied.
Now consider the field Lq = Q(ζq, a1/q), let p be a rational prime that splits
completely in Lq and let P be a prime over p. The residue field at P has p elements,
therefore (1) holds. Conversely if (1) holds for p, then p splits completely in Lq.
The Chebotarev Density Theorem indeed states that the probability that p splits
completely in a normal extension K, equals 1/[K : Q] and therefore the probability
Pa(q) is 1/q(q − 1) and
A(a) =∏q
(1− 1
q(q − 1)
).
3
Later, calculations made by D. H. Lehmer and E. Lehmer (see [35]) suggested that
in some cases the expression of A(a) was not correct and the factors of the product
expansion of A(a) corresponding to the prime divisors of a had to be replaced by
other expressions.
In 1965, C. Hooley (see [26]) used the linear sieve to prove that if the validity of
the Generalized Riemann Hypothesis is assumed for the Dedekind zeta function of
the fields Lq then the Artin’s Conjecture is true, with the corrections indicated by
Lehmer.
The main tool used by Hooley is the effective version of the Chebotarev density
Theorem valid under the assumption of the Riemann Hypothesis for the Dedekind
zeta function of K. That is
#p ≤ x | p splits completely in K =1
nK
li(x) + O(x1/2(log x+ log d1/nK
K )),
where nK = [K : Q] and dK is the discriminant.
This version of the Chebotarev Density Theorem has been for a long time the
only effective one available until 1977 when J. C. Lagarias and A. M. Odlyzko proved
a version of the Theorem valid with the condition (see [29]):√log x
nK
maxd
1/nK
K , log dK
.
For a Kummer’s extension of the type Lq, this is equivalent to q < log1/6 x.
Such a discovery, unfortunately, does not allow one to eliminate the use of the
Riemann Hypothesis on the proof of the Theorem of Hooley, however it gives a
uniform result for q < log1/6 x.
In 1984 R. Gupta and R. Murty (see [15]) published the first result in which the
validity of the Artin Conjecture is established for at least one value of a. Indeed,
4
they constructed a set of 13 numbers for which at least one is primitive root for a
number of primes p up to x which is xlog2 x
. This result was later sharpened by
Heath-Brown (see [21]) to a set of 3 elements.
The idea of Gupta and Murty also allowed them to deal with the analogous
statement of the Artin Conjecture for rational points on Elliptic Curves (see. [17]).
This is the Lang-Trotter Conjecture. From this they were led to consider a high-rank
version of the Artin Conjecture.
Given a1, . . . , ar ∈ Z, we say that a1, . . . , ar are multiplicatively independent
if, whenever there are integers n1, . . . , nr such that
an11 · · · anr
r = 1,
we have n1 = n2 = · · · = nr = 0.
It makes sense to ask if
〈a1, . . . , ar mod p〉 = F∗p (2)
for infinitely many primes p and to speculate whether the density of such primes can
be calculated. It is necessary to express the condition for a prime q to divide the
index of the group generated by a1, . . . , ar in terms of splitting conditions on some
fields. The natural generalization of Artin’s original idea is in:
Theorem 1 Let 〈a1, . . . , ar〉 be the subgroup of F∗p generated by the multiplicatively
independent a1, . . . , ar. For any prime q
q∣∣∣[F∗
p : 〈a1, . . . , ar〉] ⇐⇒ p splits completely in Q(ζq, a
1/q1 , . . . , a1/q
r
).2
This result and the consequent application of the Chebotarev Density Theorem
suggests that the density of primes for which (2) holds equals∏q
(1− 1
qr(q − 1)
). (3)
5
In Chapter 1 we prove that if the Generalized Riemann Hypothesis holds for the
fields in Theorem 1, then
Na1,...,ar(x) = #p ≤ x | 〈a1, . . . , ar〉 = F∗
p
∼ Aa1,...,arπ(x). (4)
where the constant Aa1,...,ar equals the product in (3), up to finitely many factors.
The complete formulas, with the analogous corrections of those suggested by Lehmer
for the Artin Conjecture, are worked out in Section 2.1 by the use of some properties
of Kummer’s extension.
The proof follows the original one of Hooley but now the estimate for the number
of primes for which there is a large prime divisor of the index is made using a Lemma
due to C.R. Matthews which is an application of the pigeon-hole principle.
The new parameter given by the rank, suggests to take r as a function of x and
try to adapt the proof to obtain a result independent of the Riemann Hypothesis.
This is done in Section 3.1 and the conclusion is that for a positive density of primes
p, F∗p can be generated by about log p multiplicatively independent integers.
The main obstacle comes from those primes for which the index
[F∗p : 〈a1, . . . , ar〉]
has some prime divisor in the interval [log1/6 x, log2 x].
The range [log1/6 x, log x] is dealt by using a version of Chebotarev Density Theo-
rem for the field Q(ζl, 21/l) valid for a range of l up to log x/(log log x)2 which is proven
in Chapter 2. Such a proof uses properties of the single non-Abelian L-function of
Q(ζl, 21/l), and is of course stronger than the one of Lagarias and Odlyzko of [29].
This establishes a Conjecture of H. Zassenhaus of 1969 (see [4]).
In Section 3.3, we work out the bound of r ≥ log2 p for a set of density one of
primes p for which F∗p is generated by r elements. Such a result is stronger than
6
the one that was known as a consequence of the work of Burgess and Elliot (see
Proposition 1.11) and uses the Large Sieve Inequality.
The Lemma of Matthews used in the proof of the asymptotic formula in (4) allows
one to conclude that for almost all primes p the index
[F∗p : 〈a1, . . . , ar〉] ≥
pr/(r+1)
log p.
In Appendix A, we improve such a lower bound to
[F∗p : 〈a1, . . . , ar〉] ≥ pr/(r+1) explogδ p. (5)
This is done by deducing an upper bound for the number of primes p for which p− 1
has a divisor in the range (xh, xh explogδ p) which is due to Murty and Erdos (see
[14]) and proven here with the uniformity conditions that allow estimates of the type
(5) uniform with respect to r.
Next we take into consideration the problem of determining an asymptotic formula
for the number of primes for which two given numbers (or more in general s given
numbers) are simultaneously primitive roots. An heuristic argument similar to Artin’s
suggests a density
δ =∏
q prime
(1− 2q − 1
q2(q − 1)
),
and again this is proven to be the right one up to finitely many factors. Complete
formulas are worked out in the case where the given numbers are primes. We later
discovered that a general version of this statement has been proven by K. R. Matthews
in his Ph.D. Thesis (see [36]). However, our proof is different and by the use of a
Tauberian Theorem, we get a better uniform error term.
This result has, as an application, a uniform asymptotic formula for the number
of primes for which the least prime primitive root is less than a parameter y. Such a
formula has applications to the problem of the distribution of least primitive roots.
7
In Appendix B, we consider the problem of the exponent of the class group of
imaginary quadratic fields.
If e(d) is the exponent of the ideal classgroup of the imaginary quadratic fields
Q(√−d), the Iwasawa Conjecture states that
limd→+∞
e(d) = +∞.
In 1972, D.W. Boyd and H. Kisilevsky (see [3]) proved that if the Extended Riemann
Hypothesis holds for certain Dirichlets L-functions, then the Iwasawa Conjecture is
true.
The proof consists on noticing a link between the least prime p for which −d is a
quadratic residue and e(d) (this is pe(d) d) and then use the Riemann Hypothesis
to prove that p log2 d. This argument establishes the bound
e(d) log d
log log d. (6)
We prove unconditionally that (6) holds for a set of discriminants of density one,
by calculating uniform asymptotic formulas for the number of integers (resp. square-
free integers) d < x for which the least prime p with(−dp
)= 1 is smaller than s.
8
1 ON HOOLEY’S THEOREM
1.1 A generalization of Hooley’s Theorem
Suppose a1, . . . , ar are multiplicatively independent integers and let Γ be the subgroup
of Q× generated by a1, . . . , ar. For all but finitely many primes p, it makes sense to
consider the reduction of Γ modulo p which we indicate by Γp which can be viewed
as a subgroup of F∗p.
In the case r = 1, Hooley has shown that if the generalized Riemann Hypothesis
holds for the Dedekind zeta function of the fields Q(ζl, a1/l1 ), with l prime, then the
set of primes p for which F∗p = Γp has non zero density (see [26]).
We will consider the following generalization first introduced by R. Gupta and R.
Murty in [15].
Theorem 1.1 Let Γ be as above, nm = [Q(ζm, a1/m1 , . . . , a1/m
r ) : Q] and let
δΓ =∞∑
m=1
µ(m)
nm
if the Generalized Riemann Hypothesis holds for the Dedekind zeta function of the
fields Q(ζl, a1/l1 ), l prime, then
NΓ(x) = #p ≤ x|F∗p = Γp ∼ δΓ
x
log x.
Remark: a) Note that
nm ≥ [Q(ζm, a1/m1 ) : Q] φ(m)m, (1)
therefore δΓ is a convergent series and thus a well defined number. We will prove in
the second section that δΓ 6= 0.
9
b) Theorem 1.1 can also be proven on the weaker assumption that there exists a ∈ Γ
with the property that all the Dedekind zeta functions of the fields Q(ζl, a1/l) (l large
prime) have no zeroes in the region
σ > 1− 1
r + 1.
Proof: Let us assume r > 1. The first steps of the proof follow the original idea
of Hooley who considered the following functions:
NΓ(x, y) = #p ≤ x | ∀l, l ≤ y, l 6 | [F∗p : Γp],
MΓ(x, y, z) = #p ≤ x | ∃l, y ≤ l ≤ z, l| [F∗p : Γp],
MΓ(x, z) = #p ≤ x | ∃l, l ≥ z, l| [F∗p : Γp],
where y = 16r+8
log log x and z = x1/(r+1) log x.
Clearly,
NΓ(x, y) ≥ NΓ(x) ≥ NΓ(x, y)−MΓ(x, y, z)−MΓ(x, z), (2)
and establishing the following:
a) NΓ(x, y) = δΓx
log x+ o( x
log x);
b) MΓ(x, y, z) = o( xlog x
);
c) MΓ(x, z) = o( xlog x
),
the Theorem would be proven.
In his original work, Hooley used the GRH to treat both the main term NΓ(x, y)
and the term MΓ(x, y, z). In this proof we will show that a choice of y = 16r+8
log log x
enables to remove the GRH from the treatment of the main term. This is a key
element for subsequent applications.
10
a) By the inclusion-exclusion formula,
NΓ(x, y) =∗∑m
µ(m)πm(x)
where µ is the Mobius function, the upper ∗ means that the sum is extended to all
the integers m whose prime divisors are distinct and less than y (note that this forces
m ≤ ∏q<y q = eθ(y) < e2y, the last inequality being implied by the Prime Number
Theorem) and
πm(x) = #p ≤ x | ∀q, q|m, q| [F∗p,Γp].
Now recall that
q | [F∗p : Γp] ⇐⇒ p splits completely in Q(ζq, a
1/q1 , . . . , a1/q
r ),
and if a prime splits completely in two fields then it does also in their compositum.
Hence if Lm = Q(ζm, a1/m1 , . . . , a1/m
r ), we have
πm(x) = #p ≤ x | p splits completely in Lm. (3)
The result that gives an asymptotic formula for (3) and makes possible to handle
this step without the use of the GRH is the Chebotarev Density Theorem, with the
error term described in page 243 of [39]:
Lemma 1.2 (Chebotarev Density Theorem): If L is a Galois extension of Q
with discriminant dL and degree nL, then there exists an absolute constant c such that
for √log x ≥ c n
1/2L max(log |dL|, |dL|1/nL),
one has
#p ≤ x | p splits completely in L =1
nL
li(x) + O(x exp−An−1/2L
√log x)
where A is constant depending only on c.2
11
Now, let dm be the discriminant of Lm and nm its degree. The Hensel inequality
(see. page. 259 of [42]) states that
log |dm| ≤ nm
∑q|dm
log q, (4)
therefore
d1/nmm ≤
∏q|dm
q ≤ ma1 . . . ar ≤ nm ≤ log dm
since indeed in any field log d ≥ n. We can also prove the following.
Corollary 1.3 If m ≤ (log x)1
3r+4 then
πm(x) =li(x)
nm
+ O(x exp−A(log x)1/3)
for some absolute positive constant A.
Proof of Corollary 1.3: The inequality assumed form and the Hensel inequality
in (4), imply (nm ≤ mr+1):
c n1/2m log dm ≤ c n3/2
m
∑q|dm
log q ≤ m3r+4
2 ≤ (log x)1/2.
Hence, Lemma 1.2 gives
πm(x)− li(x)
nm
= O
x exp
−A( log x
nm
)1/2
= O(x exp
(−A(log x)
12− r+1
2(3r+4)
))= O
(x exp
(−A(log x)1/3
)).2
The choice made for y allows us to apply Corollary 1.3 to all the m ≤ e2y =
(log x)1
3r+4 . Using the estimate (1) for the degree nm, we get:
NΓ(x, y) =∗∑m
µ(m)(
1
nm
li(x) + O(x exp−A(log x)1/3))
=
12
=∞∑
m=1
µ(m)
nm
li(x) + O
∑m>y
1
mφ(m)li(x)
+ O(eyx exp−A(log x)1/3)
=∞∑
m=1
µ(m)
nm
x
log x+ o(li(x)) + O((log x) x exp(−A(log x)1/3))
= δΓx
log x+ o
(x
log x
).
c) To deal with the last term, we will make use of the following result due to
Matthews (see [37]):
Lemma 1.4
#p | |Γp| ≤ t = O(t1+1/r∑
i
log ai)
where the constants involved in the O symbol do not depend on t nor r, nor on
a1, . . . , ar.
Proof of Lemma 1.4: Consider the set S = an11 · . . . · anr
r | 0 ≤ ni ≤ t1/r. As
a1, . . . , ar are multiplicatively independent, the number of elements of S exceeds
([t1/r] + 1)r > t.
If p is prime such that |Γp| ≤ t, then two distinct elements of S are congruent
(modp). Hence, p divides the numerator N of
am11 · · · amr
r − 1
for some m1,m2, . . . ,mr satisfying |m1| ≤ t1/r, 1 ≤ i ≤ r.
For a fixed choice of m1,m2, . . . ,mr, the number of such primes is bounded by
logN ≤ t1/rr∑
i=1
log ai
Taking in account the number of possibilities for m1,m2, . . . ,mr, the total number of
primes p cannot exceed
O(t1+1/rr∑
i=1
log ai).
13
This completes the proof of the Lemma.2
Now note that
MΓ(x, z) ≤ #
p ≤ x | ∃ l ≥ z, l|p− 1
|Γp|
≤ #p ≤ x | |Γp| ≤
x
z
and applying Lemma 1.4 (no dependence on r is required here), we get
MΓ(x, z) = O
(x(1−1/(r+1))(1+1/r)
(log x)1+1/r
)= o
(x
log x
).
b) For the middle term we assume the GRH which allows to state the following
version of the Chebotarev Density Theorem (a proof can be found in [26] or also in
[30]):
#p ≤ x | p splits completely in Q(ζl, a1/l1 ) =
1
l(l − 1)li(x) + O(x1/2 log xl) (5)
Now, as in the main term, l|[F∗p : Γp] if and only if p splits completely in the
Kummer extension Q(ζl, a1/l1 , . . . , a1/l
r ) and thus, in particular, p splits completely in
Q(ζl, a1/l1 ). From this we get:
MΓ(x, y, z) ≤ #p ≤ x | ∃ l, y ≤ l ≤ z, p splits completely in Q(ζl, a1/l1 )
≤∑
y≤l≤z
(1
l(l − 1)li(x) + O(x1/2 log xl)
).
As∑
l≥y1
l(l−1)is the tail of a convergent sequence and
∑l<z
x12 log xl x
12+ 1
r+1 log x,
for r > 1 this yields to an estimate of the type:
MΓ(x, y, z) 1
yli(x) + O(x
12+ 1
r+1 log x) (6)
14
which is o(log x/x), and this completes the proof for r > 1.
For completeness we add the the proof of the remain case r = 1. Estimate (6)
has no meaning anymore. We have that
MΓ(x, y, x1/2 log x) ≤MΓ(x, y, x1/2/ log3 x) +MΓ(x, x1/2/ log3 x, x1/2 log x).
The first term is treated as the general case and leads to the coresponding estimate
of (6) that in this case is o(
xlog x
). For the second term we proceed as Hooley and we
note that z = x1/2 log x and
MΓ(x, x1/2/ log3 x, x1/2 log x) ∑
x1/2/ log3 x<l<x1/2 log x
π(x, l, 1)
x
log x
∑x1/2/ log3 x<l<x1/2 log x
1
l= O
(x log log x
log2 x
).
the last by the Brun-Titchmarsh Theorem and the Merten’s formula.2
15
1.2 Computation of the Densities
The density δΓ can always be expressed as an Euler product. Doing so one can prove
that the density is not zero. In this section we will calculate δΓ in the case when
ai = pi is an odd prime for any i ≥ 1, we will also be able to prove that in this
particular case
limr→∞
δΓ = 1.
The first step is to calculate the degrees of Lm over Q.
Theorem 1.5 Let p1, . . . , pr be odd primes, m a square-free integer and let
nm = [Q(ζm, p1/m1 , . . . , p1/m
r ) : Q].
Suppose (m, p1 · · · pr) = pi1 · · · pit, then nm = φ(m)mr
2α , where
α =
0 m is odd or t = 0
t if pi1 ≡ pi2 ≡ · · · ≡ pit ≡ 1 (mod 4)
t− 1 otherwise.
Proof: Fix m > 1, we may assume without loss of generality that p1 · · · pt =
(p1 · · · pr,m), we let K = Q(ζm), A = K(p1/m1 , . . . , p
1/mt ) and for any 1 ≤ i ≤ r − t,
let Bi = A(p1/mt+1 , . . . , p
1/mt+i ). We have that
nm = [Br−t : Q] = [Br−t : A][A : K][K : Q]
and clearly [K : Q] = φ(m).
Step 1): We claim that [Br−t : A] = mr−t.
Since the polynomial xm − pt+1 splits completely in B1 = A(p1/mt+1 ), we know that
[B1 : A] = md. Let q|d be a prime, then [A(p
1/qt+1) : A] = 1 or q. If it was q, we would
have q = [A(p1/qt+1) : A]|[B1 : A] = m
d, which is a contradiction since m is square-free.
16
Therefore p1/qt+1 ∈ A, which implies that pt+1 ramifies in A/Q, but, from Kummer’s
Theory, we know that the only primes that ramify in A are p1, . . . , pt and those that
divide m, and since (pt+1,m) = 1, we conclude that d = 1. Now, by induction, we
have that
[Br−t : A] = [Br−t : Br−t−1][Br−t−1 : A] = [Br−t : Br−t−1]mr−t−1,
and again, [Br−t : Br−t−1] = md
and since (pr,m) = 1, we conclude that d = 1. Hence
[Br−t : A] = mr−t.
Step 2) Let Ai = K(p1/m1 , . . . , p
1/mi ), then Ai+1 = Ai(p
1/mi+1 ), and for the same
reason as above, [Ai+1 : Ai] = me. We claim that e = 1 or 2.
Let q|e be a prime divisor and consider Ai(p1/qi+1), since m is square-free, we have that
p1/qi+1 ∈ Ai. If p
1/qi+1 ∈ K, then we would have a cyclic extension of prime degree (over
Q) Q(p1/qi+1) ⊂ K and this is only possible when q = 2. Therefore we may assume
that p1/qi+1 6∈ K, having extensions:
K ⊆ K(p1/qi+1) ⊆ Ai.
Note that Gal(Ai/K) is the direct product of cyclic groups and a general subgroup of
order q has as fixed field K((ps1 · · · psk)1/q), with 1 ≤ s1 ≤ · · · ≤ sk ≤ i−1. Therefore,
K(p1q
i+1) = K((ps1 · · · psk)1/q) and from Lemma 3 in page 160 of Cassels and Frohlich
[7], we have that there exists 0 ≤ i ≤ q − 1 such that(pi+1
(ps1 · · · psk)i
)1/q
∈ K,
and again this implies that q = 2.
Therefore, if m is odd, [Ai+1 : Ai] = m for every i, and thus [At : K] = mt.
From the Theory of Cyclotomic Fields, we know that the general quadratic sub-
field of K has the form Q(√(
−1D
)D), where D is a positive divisor of m. We gather
that if pi ≡ 1 (mod 4), 1 ≤ i ≤ t, then(−1pi
)= 1, hence
√pi ∈ K.
17
Step 3) If p1 ≡ p2 ≡ · · · ≡ pt ≡ 1 (mod 4),
then let ζm be a primitive m-th root of unity, then Gal(A1/K) is generated by σ :
p1/m1 7→ ζ2
mp1/m1 , (Note that σ(
√p1) = (σ(p
1/m1 ))m/2 = (ζ2
m)m/2p(1/m)(m/2)1 =
√p1) and
hence, |Gal(A1/K)| = [A1 : K] = m2.
Similarly Gal(Ai+1/Ai) is generated by σ : p1/mi+1 7→ ζ2
mp1/mi+1 , therefore [Ai+1 : Ai] = m
2
and [A : K] = mt
2t .
Step 4) If it exists 1 ≤ i ≤ t such that pi ≡ 3 (mod 4),
then we can suppose without loss of generality that p1 ≡ 3 (mod 4). Let us consider
Ai = K(p1/m1 ). We have that [A1 : K] = m (If not, we would have K(
√p1) = K,
but this only happens when p1 ≡ 1 (mod 4), which is a contradiction). Now consider
i > 1, and Ai = Ai−1(p1/mi ). We claim that [Ai : Ai−1] = m
2. Indeed either pi ≡ 1
(mod 4) or pi ≡ 3 (mod 4); in the first case√p1 ∈ K, in the second case
√p1pi ∈ K.
In any case, Gal(Ai/Ai−1) is always generated by σ : p1/mi 7→ ζ2
mp1/mi . Finally we get
[Ai : Ai−1] = m2
and [A : K] = mt
2t−1 .
This concludes the proof of the Theorem.2
Corollary 1.6 With the same notation of Theorem 1.5, we have
nm ≥ mrφ(m)/2min(r,ν(m)−1)
(where ν(m) is the number of distinct prime divisors of m), furthermore such a lower
bound is the best possible.2
We are now ready to express the density as an Euler product. The case r = 1 has
been dealt with by C. Hooley in [26]. He proved that:
Lemma 1.7 Let p be a prime, nm = [Q(ζm, p1/m) : Q] and let
A =∏
l prime
(1− 1
l(l − 1)
)
18
be the Artin’s constant, then we have:
∞∑m=1
µ(m)
nm
=
A if p 6≡ 1 (mod 4),
A(1 + 1
p2−p−1
)if p ≡ 1 (mod 4).
Proof: If p 6≡ 1 (mod 4), then nm = mφ(m) for every m and the result follows
from the definition of the Artin’s constant. We can therefore assume that p ≡ 1
(mod 4), having:∞∑
m=1
µ(m)
nm
= Σo + Σe,
where Σo is the sum extended to the odd values of m and Σe to the even values.
Clearly Σo = 2A and Σe = −12Σ′
e, with
Σ′e =
∞∑m=1
(m,2p)=1
µ(m)
mφ(m)+ 2
∞∑m=1
p|m,modd
µ(m)
mφ(m)=
2A+−1
p(p− 1)
∑m=1
(m,2p)=1
µ(m)
mφ(m)= 2A+
−1
p(p− 1)
2A(1− 1
p(p−1)
) = 2A− 2A
p2 − p− 1.
Finally Σo + Σe = A(1 + 1
p2−p−1
).2
The general case is similar but a little more complicated:
Theorem 1.8 Let p1, . . . , pr be odd primes, nm = [Q(ζm, p1/m1 , . . . , p1/m
r ) : Q], let
αi = pri (pi − 1)− 1 and define the r-dimensional incomplete Artin’s constant to be:
A(r) =∏
l odd prime
(1− 1
lr(l − 1)
),
then:
∞∑m=1
µ(m)
nm
= A(r)
1− 1
2r+1
[r∏
i=1
(1−
(−1
pi
)1
αi
)+
r∏i=1
(1− 1
αi
)].
19
Proof: As in the case r = 1, note that if m is odd, then nm = mrφ(m), thus we
can write:∞∑
m=1
µ(m)
nm
= A(r)− Σ
where Σ is the sum extended to the even values of m.
Let P = p1 · · · pr and P =∏r
i=1, pi≡1(4) pi, if m is an odd positive integer and Q =
(m,P ) then, by Theorem 1.5, we have
n2m =
2r mrφ(m)
2ν(Q) if Q|P
2r mrφ(m)
2ν(Q)−1 otherwise.
For any Q|P , let S(Q) = m ∈ N| (m,P ) = Q. We have that N =⋃
Q|P S(Q), and
the union is disjoint. Therefore,
Σ =∑Q|P
∑m∈S(Q)
µ(2m)
n2m
.
Now divide the set of divisors of P into two sets; the divisors of P , and its complement.
It follows that
Σ =∑Q|P
∑m∈S(Q)
µ(2m)2ν(Q)
2rmrφ(m)+∑Q|PQ|/P
∑m∈S(Q)
µ(2m)2ν(Q)−1
2rmrφ(m)=
1
2r+1
∑Q|P
2ν(Q)∑
m∈S(Q)
µ(2m)
mrφ(m)+∑Q|P
2ν(Q)∑
m∈S(Q)
µ(2m)
mrφ(m)
.The sum over m ∈ S(Q) is easy to evaluate,
∑m∈S(Q)
µ(2m)
mrφ(m)= −(−1)ν(Q)
Qrφ(Q)
∑(m,2P )=1
µ(m)
mrφ(m)= −(−1)ν(Q)
Qrφ(Q)A(r)
r∏i=1
(1− 1
αi + 1
)−1
.
Substituting we get:
Σ =−A(r)
2r+1
r∏i=1
(1− 1
αi + 1
)−1
∑Q|P
(−2)ν(Q)
Qrφ(Q)+∑Q|P
(−2)ν(Q)
Qrφ(Q)
=
20
−A(r)
2r+1
r∏i=1
(αi + 1
αi
) r∏i=1
pi≡1(4)
(1− 2
αi + 1
)+
r∏i=1
(1− 2
αi + 1
) =
−A(r)
2r+1
r∏i=1
pi≡1(4)
(1− 1
αi
) r∏i=1
pi≡3(4)
(1 +
1
αi
)+
r∏i=1
(1− 1
αi
) .The claim is therefore deduced.2
Corollary 1.9 Let aii>1 be a sequence of odd primes and let δr be the density of
the set of primes p for which F∗p is generated by a1, . . . , ar, then
limr→∞
δr = 1.2
Remark:
The method just exposed can be easily extended to any set of r multiplicatively
independent numbers which are pairwise coprime. The first step of the induction
in the general form is in [26]. It is also conceivable that for any infinite sequence of
multiplicatively independent integers (that is a sequence of integers such that ai < ai+1
and for any r, a1, . . . , ar are multiplicatively independent), one has that limr→∞ δr =
1. Not being able to provide a proof of this property here, we will include it in the
hypothesis when ever needed.
21
1.3 The Main Problem
Suppose f(p) is a monotone function of p that tends to infinity with p and let ann∈N
be an infinite sequence of multiplicatively independent integers. Let
Γf,p = 〈ai (mod p)| 1 ≤ i ≤ f(p)〉.
Question:
Does a function exist f such that, Γf,p = F∗p for almost all primes p ?
Using Theorem 1.1, we can prove:
Theorem 1.10 Let aii∈N be a sequence of multiplicatively independent integers
such that
limr→∞
δΓ = 1
(We noticed in the last section that when the ai’s are all primes this is true) sup-
pose the Generalized Riemann Hypothesis holds for the Dedekind function of the field
Q(ζl, a1/l1 ), l prime, then for any monotone function f(p) that tends to infinity, we
have that
#p ≤ | Γf,p = F∗p ∼ π(x).
Proof: Let us fix r ∈ N. For all but finitely many primes p, we have:
Γf,p ⊃ Γp = 〈a1, . . . , ar〉.
Therefore
A = #p ≤ x | Γf,p = F∗p ≥ #p ≤ x | Γp = F∗
p+ O(1).
From Theorem 1.1, we get:
limx→∞
Ax
log x
≥ δΓ.
22
Now let r tend to infinity and prove the statement.2
Our intention in the following Chapters is to prove statements of the type of
Theorem 1.10, restricting our assumptions only on the rate of growth of f . For
example it is not difficult to prove:
Proposition 1.11 Let f(p) = log2+ε p then there exists a sequence of multiplicatively
independent integers such that, for almost all primes p, Γf,p = F∗p.
Proof: This is a consequence of a Theorem of Burgess and Elliott (see [5]) on
the average of the least primitive root. They proved that if g(p) is the least primitive
root, then for large x,
π(x)−1∑p≤x
g(p) log2 x(log log x)2.
If U is the number of primes up to x for which g(p) ≥ f(p), we get that:
U log2+ε x∑
xlog x
≤p≤x
g(p) + o(π(x) log2+ε x
) π(x) log2 x(log log x)2
which is equivalent to saying that for almost all primes g(p) ≤ f(p).
Now let ai = pi be the i-th prime number, since g(p) ≤ log2+ε p, every prime that
divide g(p) is also less than f(p), therefore Γf,p contains a primitive root for almost
all primes p.2
23
2 ON THE ARTIN L-FUNCTIONS OF Q(ζl, 21/l)
2.1 Introduction
Let L = Q(ζl, 21/l) and G = Gal(L/Q). If
τ : L −→ L and ν : L −→ L
21/l 7→ ζl21/l 21/l 7→ 21/l
ζl 7→ ζl ζl 7→ ζgl
(where g is a primitive root modulo p), then G is generated by τ and ν, more precisely,
G =< τ, ν | τ l = νl−1 = 1, ν−1τν = τ g∗ >
is a presentation (here g∗ is any integer such that gg∗ ≡ 1 mod p).
Hence G is the semidirect product of a cyclic group of order l by a cyclic group of
order l−1. Note also that τ generates the Galois group of L/Q(ζl) and the subgroup
generated by ν has as fixed field, the non-Galois field K = Q(21/l).
For any t = 1, . . . , l − 1, the map
χt : G→ G, τ 7→ 1, ν 7→ e2πitl−1
is clearly a character and a quick computation shows that G has l conjugate classes
and the remaining character of G can be calculated via the orthogonality relations.
That is
χl(τaνb) =
(l − 1) If a = b = 0
0 If b 6= 0
−1 If b = 0, and a 6= 0.
Note also that χl is induced by any non-trivial character of the normal subgroup
generated by τ and if φt : ν 7→ e2πit/(l−1), t < l− 1 is a character of the subgroup 〈ν〉,
24
then indG〈ν〉φt = χt + χl.
Hence χ1, . . . , χl is a complete list of the irreducible characters of G.
Let us now take a step back and describe the concept of non-Abelian Artin L-
function. Let E/F be a Galois extension and ρ a representation of Gal(E/F ), we
define the Artin L-function of ρ to be
L(s, ρ, E/F ) =∏℘
L℘(s)
where, if ℘ does not ramify, the Artin symbol σ℘ is the conjugacy class in
Gal(E/F ) determined by the Frobenius automorphism of the residue field of any
prime of E over ℘ (note also that, if ℘ does not ramify in E, then σ℘ = 1 if and
only if ℘ splits completely in E) and L℘(s) is the characteristic polynomial of σ℘
evaluated at N(℘)−s, i.e.
L℘(s) = det(I −N(℘)−sρ(σ℘))−1
and, if ℘ is ramified, L℘(s) is the characteristic polynomial of the Frobenius element
at ℘ acting on the subspace fixed by the inertia group I℘ evaluated at N(℘)−s.
Simple arguments on the bounds of the eigenvalues of the representation show
that L(s, ρ, E/F ) converges absolutely for <(s) > 1. Since the determinant of a
matrix is the product of its eigenvalues, we also have that:
logL(s, ρ, E/F ) =∑℘, m
tr(ρ(σ℘)m)
m℘ms
Sometimes, we might indicate L(s, ρ, E/F ) by L(s, χ, E/F ) where χ is the character
of ρ.
We describe here the basic properties of L-functions. For a more complete picture,
see [33] Chapter XII.
25
PROPERTIES:
A) If ZF (s) is the usual Dedekind zeta function of the field F , then
ZF (s) = L(s, 1, E/F );
B) If χ1, χ2 are two characters of Gal(E/F ), then
L(s, χ1 + χ2, E/F ) = L(s, χ1, E/F ) + L(s, χ2, E/F );
C) If E ′ ⊃ E ⊃ F , where E ′/F is also Galois, then any character of Gal(E/F ) can
be viewed as a character of Gal(E ′/F ) (by composing Gal(E ′/F ) −→→ Gal(E/F )χ−→
C), and we have
L(s, χ, E/F ) = L(s, χ, E ′/F );
D) If E ⊃ E ′ ⊃ F then Gal(E/E ′) ⊂ Gal(E/F ), therefore any character χ of
Gal(E/E ′) induces a character Ind(χ) of Gal(E/F ) and one has:
L(s, Ind(χ), E/F ) = L(s, χ, E/E ′);
E) If E/F is an abelian extension then for every character χ, L(s, χ, E/F ) has an
extension to an entire function and verifies a functional equation;
F) If χreg is the character of the regular representation of Gal(E/F ), then
L(s, χreg, E/F ) = ZE(s).
(This is a consequence of the fact that the regular representation is induced by the
trivial character on the trivial identity subgroup which is the Galois group of L/L
therefore, D) and A) give this claim);
G) The Brauer Theorem for characters, states that any character is equal to a
sum with integer coefficients of characters induced from elementary subgroups (see
26
[46] Chapter X). By properties B), C), and D) this implies that any Artin L-function
can be written as product of powers with integer exponents of entire functions, and
therefore, any such a function is certainly meromorphic. Artin had actually conjec-
tured that these functions are always entire whenever χ does not contain the trivial
character;
H) Whenever the Galois group of an extension has the property that every char-
acter is induced by the character of an abelian subgroups (such characters are called
monomials) then the Artin Conjecture holds for such an extension. This is the case
of nilpotent extensions (as well as supersolvable extensions).
2.2 Artin L-functions of L/Q
The Galois group G of L/Q is certainly supersolvable. Thus all the Artin L-functions
of G are entire and by the properties F), D) and A), we have the following factoriza-
tion:
ZL(s) = ζ(s)
(l−2∏t=1
L(s, χt, L/Q)
)L(s, χl, L/Q)l−1.
On the other hand, if K = Q(21/l),
ZL(s) = L(s, χreg, L/K) = ZK(s)l−2∏t=1
L(s, φt, L/K) =
ZK(s)l−2∏t=1
L(s, χt, L/Q)L(s, χl, L/Q),
the last identity being obtained noticing that Ind(φt) = χt + χl and applying prop-
erties B) and D). Putting the two together, we get
ZK(s) =ZL(s)(∏l−2
t=1 L(s, χt, L/Q))L(s, χl, L/Q)l−2
=ζ(s)
(∏l−2t=1 L(s, χt, L/Q)
)L(s, χl, L/Q)l−1(∏l−2
t=1 L(s, χt, L/Q))L(s, χl, L/Q)l−2
= ζ(s)L(s, χl, L/Q).
27
Therefore the zeroes of L(s, χl, L/Q) are in particular zeroes of ZK(s) and
L(1, χl, L/Q) = Ress=1ZK(s) 6= 0
The identity also allows us to compute the functional equation for L(s, χl, L/Q).
It is indeed a classical result that if K is any number field, and
FK(s) = AsΓ(s
2
)r1
Γ(s)r2ZK(s) (1)
(where A = 2−r2d1/2K π−nK/2, r1 and r2 are respectively the number of real and complex
embeddings of K, dK is the absolute value of the discriminant of K and nK its degree
over Q) then FK(s) = FK(1− s).
In our case dK = 2l−1ll, nK = l, r1 = 1, r2 = (l − 1)/2, and
ZK(s) = ζ(s)L(s, χl, L/Q),
so we get:
FK(s) =
(l
π
)(l/2)s
Γ(s
2
)Γ(s)(l−1)/2ζ(s)L(s, χl, L/Q).
Using the fact that the value of πs/2Γ(
s2
)ζ(s) does not change if we substitute s
with 1− s, we get that if
G(s) =
(l
π
)(l/2)s
πs/2Γ(s)l−12 L(s, χl, L/Q) (2)
then G(s) = G(1− s), which is the functional equation.
An asymmetric functional equation can also be deduced using the formula:
Γ(s)Γ(1− s) =π
sin πs;
which is
L(1− s, χl, L/Q) =ll(s−1/2)
π(l−1)s(sin πs)(l−1)/2Γ(s)l−1L(s, χl, L/Q) (3)
28
These results can be used to determine all the zeroes of L(s, χl, L/Q) outside of the
critical strip. Indeed L(s, χl, L/Q) has just a zero of order l−12
for s = 0,−1,−2, . . .
and is non-vanishing elsewhere (outside the critical strip).
We conclude this Section with a classical general result that we will use later (this
result can be found in [31], in that version, though, are missing all the uniformity
conditions which are necessary for subsequent applications).
Lemma 2.1 Let K be a number field, n = [K : Q], d the absolute value of the
discriminant and let ZK(s) be its Dedekind zeta function. There exists a positive
absolute numerical constant c1 such that in the region
σ ≥ 1− c1log d(t+ 2)n
, t ≥ 0
ZK(s) has no zeroes.
Proof of Lemma 2.1: We will follow the classical proof for the Riemann Zeta
function (See. [6] §13). Let HK(s) = 12s(s − 1)FK(s) where FK(s) has been defined
in (1). HK(s) is an integral function of order 1, verifies HK(1− s) = HK(s) = HK(s)
and admits the following Weierstrass product expansion:
HK(s) = ea+bs∏ρ
(1− s
ρ
)es/ρ (4)
where the product is extended to all the non-trivial zeroes of ZK(s).
Taking the logarithmic derivative and using the functional equation, we get
H ′K
HK
(s) = b+∑ρ
(1
s− ρ+
1
ρ
)= −b−
∑ρ
(1
1− s− ρ+
1
ρ
)= −H
′K
HK
(1− s).
Since, if ρ is a root then also 1− ρ is, we deduce that
<(b+
1
2
∑ρ
(1
ρ+
1
ρ
))= 0,
29
therefore
<(H ′
K
HK
(s)
)=∑ρ
<(
1
s− ρ
).
Substituting inside the real part of the logarithmic derivative of (1), we have the
identity:
∑ρ
<(
1
s− ρ
)= <
(1
s+
1
s− 1+ logA+
r12
Γ′(s/2)
Γ(s/2)+ r2
Γ′(s)
Γ(s)+Z ′
K(s)
ZK(s)
). (5)
Now consider this expression for s = σ, σ + it, σ + 2it, 1 < σ ≤ 2, t ≥ 0. Since
log(A) log d and since
<(
1
s− ρ
)=
σ − β
|s− ρ|2> 0,
there exist three absolute positive constants c2, c3, c4 such that if we take t = γ to be
the ordinate of the zero ρ = β + iγ, then
−<Z′K(σ)
ZK(σ)<
1
σ − 1+ c2 log(d);
−<Z′K(σ + it)
ZK(σ + it)< c3 log(d(t+ 2)n)− 1
σ − β;
−<Z′K(σ + 2it)
ZK(σ + 2it)< c4 log(d(t+ 2)n)
because of the Stirling formula for the Gamma function. Finally the standard in-
equality
3
[−Z
′K(σ)
ZK(σ)
]+ 4
[−<Z
′K(σ + it)
ZK(σ + it)
]+
[−<Z
′K(σ + 2it)
ZK(σ + 2it)
]≥ 0
implies4
σ − β<
3
σ − 1+ c5 log(d(t+ 2)n).
A choice of σ = 1 + δlog(d(t+2)n)
yields, for an opportune δ
β < 1− c1log(d(t+ 2)n)
which is equivalent to the statement.2
30
2.3 On the non-Abelian L-function of Q(ζl, 21/l)
Just for this Section, we will use the notation L(s) = L(s, χl, L/Q), the main goal of
this Section is to prove the following:
Theorem 2.2 With the same notations as above, there exists a positive absolute
constant A such that uniformly
π(x, χl) =∑p≤x
χl(σp) xl exp
−A√
log x
l
.Proof: In the spirit of the classical Prime Number Theorem (See. Davenport
[6]), if we define
Λl(n) =
χl(σp) log p if n is a power of a prime p
0 otherwise,
then it is sufficient to prove that
ψ(x, χl) =∑n≤x
Λl(n) xl exp
−A√
log x
l
.for some absolute positive constant A.
We will need some lemmas.
Lemma 2.3 Let N(T, χl) be the number of zeroes σ + it of L(s, χl, L/Q) such that
0 ≤ σ ≤ 1 and 0 < t ≤ T then if dK is the absolute value of the discriminant of
K = Q(21/l),
N(T, χl) = (l − 1)T
2πlog
T
2π− (l − 1)
T
2π+
(log dK
2π
)T + O(log dKT
l).
31
Proof of the Lemma: If NK(T ) and N(T ) are respectively the number of
zeroes of ZK(s) and ζ(s) in the region in question, then we have that N(T, χl) =
NK(T )−N(T ) and since, from the classical theory, we know that
N(T ) =T
2πlog
T
2π− T
2π+ O(log T ).
It is enough to show that
NK(T ) = lT
2πlog
T
2π− l
T
2π+
(log dK
2π
)T + O(log dKT
l).
Also, in the same way as in the classical result, we can write
4πNK(T ) = =(∫
R
H ′K
Hk
(s)ds
)(6)
where HK(s) is the function defined during the proof of the lemma in the last Section
and R is the rectangle, described counterclockwise, having as vertices:
5/2− iT, 5/2 + iT, −3/2 + iT, −3/2− iT .
Since HK(s) = 12s(s − 1)FK , by the residue theorem, we can write that (6) is equal
to
=(∫
R
(1
s+
1
s− 1+F ′
K
FK
(s)
)ds
)= 4π + =
(∫R
F ′K
FK
(s)ds
)(7)
If L denotes the line from 5/2 to 5/2+iT and then to 12+iT , then using the functional
equation, we quickly get that (7) equals:
4π + 4=∫L
F ′K
FK
(s)ds = 4π + 4T logA+ 4=∫L
(r1/2
Γ′(s/2)
Γ(s/2)+ r2
Γ′(s)
Γ(s)+Z ′
K(s)
ZK(s)
)ds
and, by the Stirling formula we know that
=(∫
L
Γ′(s/2)
Γ(s/2)ds
)= 2= log Γ
(1
4+ i
T
2
)= T log
T
2− T − π
4−O
(1
T
),
and
=(∫
L
Γ′(s)
Γ(s)ds
)= = log Γ
(1
2+ iT
)= T log T − T + O
(1
T
).
32
This yields
NK(T ) = lT
2πlog
T
2π− l
T
2π+
(log dK
2π
)T +
8− r18
+ O
(l
T
)+ 4=
(∫L
Z ′K(s)
ZK(s)ds
).
The problem amounts to proving that
=(∫
L
Z ′K(s)
ZK(s)ds
)= O(log dKT
l). (8)
Since ZK(s) is real on the reals, we have that ∆L arg(ZK(s)) = argZK(12
+ iT ). As
in the classical case:
Lemma 2.4 For all T > 0, we have that
∑ρ
1
1 + (T − γ)2= O(log dK(T + 2)l)
where ρ runs over all the non-trivial zeroes of ZK(s).
Proof of Lemma 2.4: From the same argument used in the proof of Lemma 2.1,
we know that for 1 < σ ≤ 2 and t > 0, there exists an absolute positive constant c0
such that
−<Z′K(s)
ZK(s)< c0 log(dK(t+ 2)l)−
∑ρ
< 1
σ − β.
Since for a choice of s = 2 + iT , we have∣∣∣∣∣Z ′K
ZK
(2 + iT )
∣∣∣∣∣ log dK ,
we obtain ∑ρ
< 1
σ − β< c1 log(dK(T + 2)l).
Finally note that
< 1
σ − β=
2− β
(2− β)2 + (T − γ)2 1
1 + (T − γ)2
and prove the Lemma.2
As in the classical case we have the following implications:
33
Corollary 2.5 For T ≥ 2, we have that
a) N(T + 1)−N(T − 1) = O(log dKTn);
b)∑
ρ,|γ−T |>1
1
(T − γ)2= O(log dKT
n).2
Now we are ready to prove (8). Take the identity (5) for s = σ + iT and s = 2 + iT ,
subtract and get:
Z ′K(σ + iT )
ZK(σ + iT )=Z ′
K(2 + iT )
ZK(2 + iT )+ O(log T l) +
∑ρ
(1
σ + iT − ρ− 1
2 + iT − ρ
) (9)
O(log dKTl) +
∑ρ,|T−γ|<1
1
σ + iT − ρ,
the last estimate because for |T − γ| > 1,∣∣∣∣∣ 1
σ + iT − ρ− 1
2 + iT − ρ
∣∣∣∣∣ ≤ 3
|γ − T |,
and ∑ρ,|T−γ|<1
1
2 + iT − ρ≤ N(T + 1)−N(T − 1) = O(log dKT
n).
Finally,
=(∫
L
Z ′K(s)
ZK(s)ds
)= =
(−∫ 2+iT
12+iT
Z ′K(σ + iT )
ZK(σ + iT )dσ
)+ =
(∫ 2+iT
2
Z ′K(s)
ZK(s)ds
).
By (9), the absolute value of the second integral is than
∑ρ,|T−γ|<1
∣∣∣∣∣∫ 2+iT
12+iT
=(σ + iT − ρ)−1
∣∣∣∣∣ dσ + O(log dKTn) =
∑ρ,|T−γ|<1
|arg(σ + iT − ρ)|2+iT12+iT
≤ π (N(T + 1)−N(T − 1)) + O(log dKTn),
while the first integral is in absolute value
|log (|ZK(2 + iT )|/|ZK(2)|)|
which is O(l).2
34
Lemma 2.6 Let x be an integer and 2 ≤ T < x, then
ψ(x, χl) = −∑|γ|≤T
xρ
ρ+ O
(xl
T(log lxT )2 + l2 log2 l
). (10)
where the sum is extended over all those zeroes ρ whose imaginary part γ is in absolute
value less or equal than T
Proof of Lemma 2.6: As in the classical case of the zeta function, ψ(x, χl) is
the sum of the coefficients of the logarithmic derivative, more precisely, if c > 1 and
T is large, then the Lemma in §17 of [6] gives that if:
J(x, T ) =1
2πi
∫ c+iT
c−iT
[−L
′(s)
L(s)
]xs
sds,
then
ψ(x, χl) = J(x, T )+O
∞∑n=1n6=x
(Λl(n)
(x
n
)c 1
T | log(x/n)|
)+cΛl(x)
T
+O(l log x) (11)
If we choose c = 1 + 1/ log x (xc = ex) and we treat the four ranges separately:
(n ≤ 34x, n ≥ 5
4x) (3
4x < n < x− 2) (x− 2 ≤ n ≤ x+ 2) (x+ 2 < n < 5
4x).
For the values of n, the first range, we have that |log(x/n)| 1 therefore the
contribution of these terms in the sum in(11) is
x
T
[−L
′(c)
L(c)
] x
Tl log l, (12)
where we just noticed that for c > 1
−L′(c)
L(c)= −
(Z ′
K(c)
ZK(c)− ζ ′(c)
ζ(c)
)=
1
c− 1− 1
c− 1+ O(log dK) = O(l log l),
and that dK = 2l−1ll.
35
For the values of n, the second range, set [x]− n = r, and note that 1 ≤ r ≤ x/4
therefore
log(x
n
)= log
(x
[x]− r
)≥∣∣∣∣∣log
(1− r
[x]
)∣∣∣∣∣ ≥ r
2x.
We gather that the contribution of these terms in (11) is
x
T
∑1≤r≤x/4
Λl(x− r)r−1 xl log x
T
∑1≤r≤x/4
r−1 xl log2 x
T(13)
Analogously, for the values of n in the fourth range, set n − [x] = r′ (now 2 ≤
r′ < x4
+ 1) and thus ∣∣∣∣log(x
n
)∣∣∣∣ = − log(1− n− x
n
) r
x
and the contribution of these terms in (11) is
xl log2 x
T. (14)
Finally for the (at most five) values of n in the third range, we have a contribution
which is l log x. Putting this together with the estimates in (12), (13) and (14),
we get:
ψ(x, χl) = J(x, T ) + O
(xl
T(log2 x+ log l)
). (15)
Now we replace the vertical segment of integration by the other three sides of the
rectangle with vertices
c− iT, c+ iT, −U + iT, −U − iT
where U is a large half integer (i.e. U = m/2, m odd integer). If T 6= γ for any zero
ρ = β + iγ of L(s) is the critical strip, the residue Theorem gives
ψ(x, χl) = −∑|γ|<T
xρ
ρ− Res
s = 0
(xs
s
L′(s)
L(s)
)− l − 1
2
U∑m=1
x−m
−m+
+O
(xl
T(log2 x+ log l)
)+ (16)
+1
2πi
(∫ −U−iT
c−iT+∫ −U+iT
−U−iT+∫ c+iT
−U−iT
)− L′(s)
L(s)
xs
s
36
since by the functional equation in (2), the integrand in J(x, T ) has simple poles at
s = ρ, poles of order (l − 1)/2 at s = −m (m ≥ 1), and an extra pole of order 2 at
s = 0.
The residue at s = 0 can be estimated as follows: Since L(s) = Zk(s)/ζ(s), and
since xs/s = 1/s+ log x+ . . ., we have
Res
s = 0
(xs
s
L′(s)
L(s)
)=
Res
s = 0
(1
s
Z ′K(s)
ZK(s)
)+ O(log x). (17)
From the functional equation in (1) and the Weierstrass product expansion in (4) we
get that
Res
s = 0
(1
s
Z ′K(s)
ZK(s)
)= b− logA+ 1− (l/2)
Γ′(1)
Γ(1)= b+ O(l log l). (18)
If we substitute s = 2 in (4), use the functional equation again and note that
Z ′K(2)/ZK(2) l, we deduce that
b =∑ρ
(1
2− ρ+
1
ρ
)+ O(l log l). (19)
For the terms of this series with |γ| ≥ 1, we have
∑|γ|≥1
∣∣∣∣∣ 1
2− ρ+
1
ρ
∣∣∣∣∣ = 2∑|γ|≥1
1
|ρ(2− ρ)|∑ρ
1
|2− ρ|2 l log l. (20)
The last sum being estimated as O(log dk) using Lemma 2.4 with t = 0. The same
estimate applies to ∑|γ|<1
1
2− ρ,
since for |γ| < 1 we have |2− ρ| |2− ρ|2.
Finally, for |γ| < 1, we know that c8/ log dK < β < 1− c8/ log dK , therefore
ρ−1 log dK ,
37
and being the number of zeroes in question log dK by Lemma 2.3 with T = 1, we
have ∑|γ|<1
1
ρ= O(log2 dK) = O(l2 log2 l). (21)
Putting together the estimates (17), (18), (19), (20) and (21) we gather
Res
s = 0
(xs
s
L′(s)
L(s)
)= O(l2 log2 l) + O(log x). (22)
From Corollary 2.5, we see that the number of non-trivial zeroes ρ of ZK(s)
for which |γ − T | < 1 is O(l log lT ), thus the differences of the ordinates of these
zeroes cannot be all o(1/(l log lT )). Hence, we can choose T (varying it by a bounded
amount, if necessary) so that |γ−T | (l log lT )−1 for all the zeroes ρ. This allows us
to determine a good bound for −L′(s)/L(s) for s = σ+ iT , T large and −1 ≤ σ ≤ 2,
that is
L′(s)
L(s)=Z ′(s)
Z(s)− ζ ′(s)
ζ(s)=
∑ρ,|γ−T |<1
1
s− ρ+ O(log dKT
l) l log2 lT (23)
where we have used (9), the fact that by our choice of T we have |γ − T |
(log dKTl)−1 for all ρ and the fact that the number of summands is here log dKT
l.
To obtain a bound for σ ≤ −1, we use the asymmetric functional equation (3)
whose logarithmic derivative is
−L′(1− s)
L(1− s)= l log l +
l − 1
2
(2 log π + π cotπs+
Γ′(s)
Γ(s)
)+L′(s)
L(s). (24)
We know that cot πs is bounded if |s−m| ≥ 1/3, that is if
|(1− s) + (m+ 1)| ≥ 1
3.
If 1 − σ ≤ −1, then Γ′(s)/Γ(s) = O(log 2|1 − s|) by the Stirling formula, while the
last term (24) is O(l). Thus
L′(s)
L(s) log 2|s|+ l log l (σ ≤ 1), (25)
38
provided that circles of radius 1/3 around the trivial zeroes s = −m of L(s) are
excluded.
Using (23) and (25) we have∫ c±iT
−U±iT l log2 lT
T
∫ c
−∞xσdσ xl log2 lT
T log x,
while (25) gives∫ −U+iT
−U−iT l log2 lU
U
∫ T
−TxUdt T l log2 lU
UxU= o(1), (for U →∞).
Inserting these estimates in (16) we get the wanted claim.2
Proof of Theorem 2.2: The zero-free region proved for ZK(s) in Lemma 2.1
holds also for L(s), therefore, if ρ = β + iγ is a zero of L(s) with γ < T < x we have
that
β < 1− c
l log lT
where c is an absolute positive constant.
We gather that
|xρ| = xβ < x exp
(−c log x
l log lT
). (26)
The sum∑
γ<T1ρ
extended over all the zeroes with |γ| > 1 can be estimated by
partial summation as follows
∑|γ|<T
1
γ=∫ T
0t−1dNl(t) =
Nl(T )
T+∫ T
0t−2Nl(t)dt
log T (l log T + log dK) l2 log2 lT.
The same sum over the zeroes ρ with |γ| ≤ 1 is O(l2 log2 l) as we noted in (21).
Putting these two facts and (26) together with Lemma 2.6 we get
ψ(x, χl) xl2 log2 lT exp
(−c log x
l log lT
)+xl
T(log lxT )2. (27)
39
We minimize it by choosing T such that
l log2 lT = log x,
and we get that (27) is
xl exp
−c/2√
log x
l
which by partial summation is equivalent to statement.2
Remark: If we assume the strong Hypothesis that for any prime l, the Dedekind
zeta function ZK(s) has the zero-free region
σ > 1− c
log T, T ≥ 0
then, using exactly the same method we would be able to prove that uniformly for
l < x
π(x, χl) xl exp(−c√
log x). (28)
2.4 An Application to Chebotarev Density Theorem
In this Section we apply Theorem 2.2 to the Chebotarev Density Theorem, obtaining
for the special case of Q(ζl, 21/l) a stronger result than Lemma 1.2. This will be used
later in Theorem 3.1, which is actually a motivation for such a result.
Theorem 2.7 There exists a constant B such that uniformly for all l with
l <log x
B(log log x)2,
we have
P (x, l) = #p ≤ x | p splits completely in Q(ζl, 21/l)
=1
l(l − 1)li(x) + O
(x exp
(−Al−1/2
√log x
)),
where A is an absolute positive constant.
40
Proof Let χG be the character of the regular representation. That is:
χG(x) =
|G| = l(l − 1) if x = 1,
0 otherwise.
We have that
1
l(l − 1)
∑p≤x
χG(σp) = #p ≤ x | σp is trivial = #p ≤ x | p splits completely in L.
On the other hand, χG = χ1 + . . .+χl−1 +(l−1)χl is the canonical decomposition
of the regular character, therefore:
1
l(l − 1)
∑p≤x
χG(σp) =1
l(l − 1)
∑p≤x
l−1∑i=1
χi(σp) +1
l
∑p≤x
χl(σp).
The orthogonality relations for the characters of the subgroup H < G give:
1
l − 1
l−1∑i=1
χi(h) =
1 if h = 1,
0 otherwise.
Therefore, the first l− 1 characters count the number of primes up to x such that
their Artin symbol is trivial modulo H. These are the primes that split completely
in the Cyclotomic field Q(ζl) (whose Galois group is isomorphic to H). Finally, if
π(x, l, 1) = p ≤ x | p ≡ 1 mod l, then
P (x, l) =1
l(π(x, l, 1) + π(x, χl)) .
The Siegel-Walfisz Theorem (see [6] in §22) states that given any positive constant
C, if l ≤ (log x)C , one has
π(x, l, 1) = #p ≤ x | p ≡ 1 mod l =1
l − 1li(x) +O(x exp(−A
√log x))
for some constant A = A(C), uniformly in l.
41
This result with C = 2 and Theorem 2.2 gives the wanted claim.2
We conclude with the following statement whose proof is a consequence of the
Remark at the end of the last Section.
Theorem 2.8 Assuming the strong Hypothesis that for any prime l and for any non-
trivial zero β + iγ of the Dedekind zeta function ZK(s), β < 1− clog γ
, then given any
positive constant C, uniformly for l ≤ (log x)C, we have
P (x, l) =1
l(l − 1)li(x) + O
(x exp
(−A
√log x
)),
for some constant A = A(C).2
42
3 ON THE NUMBER OF PRIMES GENERAT-
ING F∗p
3.1 Extending Hooley’s Method
In this Section we will extend the ideas illustrated in Chapter 1 proving the following
Theorem 3.1 Suppose Γp is the subgroup of F∗p generated by the classes of the first
log p primes, let
N(x) = #p ≤ x | Γp = F∗p
then
limx→∞
N(x)x
log x
≥ 1− log 2.
Proof: If we assume p ≥ x1/2, then, for every x,
Γp ⊇ Γr,p = 〈p1, . . . , pr〉, with r = [1
2log x]
and
N(x) ≥ N(x) = #p ≤ x | Γr,p = F∗p. (1)
Now, as in the standard Hooley’s case, we define for given η1 and η2:
N(x, η1) = #p ≤ x | ∀l, l ≤ η1, l 6 |[F∗p : Γr,p];
M(x, η1, η2) = #p ≤ x | ∃l, η1 ≤ l ≤ η2, l|[F∗p : Γr,p];
M(x, η2) = #p ≤ x | ∃l, l ≥ η2, l|[F∗p : Γr,p],
and clearly
N(x) ≥ N(x, ξ1)−M(x, ξ1, ξ2)−M(x, ξ2, ξ3)−M(x, ξ3) (2)
where ξ1 = 1/4√
log log x, ξ2 = log xB(log log x)2
and ξ3 = (log2 x)(log log x)2 and B is a
fixed positive number to be chosen later.
43
• The last term of (2) can be treated as in Theorem 1.1 using Lemma 1.4. We
have that
M(x, ξ3) ≤ #
p ≤ x | |Γr,p| ≤
x
(log2 x)(log log x)2
and since∑
i log pi = O(pr) = O(r log r), Lemma 1.4 gives
M(x, ξ3) x
(log2 x)(log log x)2
(x
(log2 x)(log log x)2
)1/r
r log r
x
log x
1
log log x. (3)
• To handle the third term of (2), we will make use of the already quoted Siegel-
Walfisz Theorem, which states that given any positive constant C, then if l ≤
(log x)C , one has
π(x, l, 1) = #p ≤ x | p ≡ 1 mod l =1
l − 1li(x) + O(x exp(−A
√log x)) (4)
for some constant A = A(C), uniformly in l.
This result yields to,
M(x, ξ2, ξ3) ≤ #p ≤ x | ∃l, ξ2 < l < ξ3, p ≡ 1 mod l ≤∑
ξ2<l<ξ3
π(x, l, 1)
=∑
ξ2<l<ξ3
(1
l − 1li(x) + O(x exp(−A
√log x))
)(5)
where we have chosen C = 3 say.
Now recall the Merten’s Theorem that states that for any two positive numbers
a and b, ∑a<l<b
1
l= log
(log b
log a
)+ O
(1
log b
).
It follows that:
∑a<l<b
1
l − 1=
∑a<l<b
1
l+
∑a<l<b
1
l(l − 1)= log
(log b
log a
)+ O
(1
log b
). (6)
44
Using this result in (5), we get
M(x, ξ2, ξ3) ≤ li(x)
(log
(log ξ3log ξ2
)+ O((log ξ2)
−1)
)+ O
(ξ3x exp−c
√log x
)
≤ x
log x
(log 2 + log
(1 + log log log x/ log log x
1− (logB + 2 log log log x)/ log log x
))+ o
(x
log x
)
=x
log xlog 2 + o
(x
log x
)(7)
• Theorem 2.7 in Chapter 2 is the ingredient to the estimate of the second term
of (2). Indeed, l < ξ2 yields to
M(x, ξ1, ξ2, ) ≤∑
ξ1<l<ξ2
#p ≤ x | p splits completely in Q(ζl, 21/l)
=∑
ξ1<l<ξ2
(1
l(l − 1)li(x) + O(xl exp
(−Al−1/2
√log x
))
1
ξ1li(x) + xξ2
2 exp(−Aξ−1/2
2
√log x
)
1
ξ1li(x) +
x log2 x
(log log x)4 logAB/2 x= o
(x
log x
), (8)
where B has been chosen to be larger than 6/A, say.
• To treat the main term of (2), let us set t = [(log r)1/2], and note that if
N0(x, ξ1) = #p ≤ x | ∀l, l ≤ ξ1, l 6 |[F∗p : Γt,p],
then N(x, ξ1) ≥ N0(x, ξ1), and
N(x, ξ1) =∗∑m
µ(m)πm(x),
where again the sum is extended to all the square free integers m whose prime
divisors are less then ξ1 (Note m ≤ eξ1), and
πm(x) = #p ≤ x | p splits completely in Q(ζm, p1/m1 , . . . , p
1/mt ).
45
Finally, the Hensel inequality (see (4) of Section 1.1) gives
c n1/2m log dm ≤ m
32t+1(logm+
∑i≤t
log pi) ≤ m2t ≤ e2ξ1t ≤√
log x.
Therefore the Chebotarev Density Theorem (see Lemma 1.2 of Section 1.1)
gives
N0(x, ξ1) =∗∑m
µ(m)
1
nm
li(x) + O
x exp
−A√
log x
nm
=∞∑
m=1
µ(m)
nm
x
log x+ O
∑m>ξ1
1
mt+1li(x)
+ O
(eξ1x exp
(−A
√log x
(log x)1/2
))
= δΓx
log x+ O
(1
ξ1
x
log x
), (9)
where δΓ is as in Section 1.1 and where we used the fact that in this range of
m, nm ≤ e2tξ1 ≤√
log x.
Putting together the estimates (3), (7), (8) and (9) and using (1) and (2), we get
N(x) ≥ N0(x, ξ1)−M(x, ξ1, ξ2)−M(x, ξ2, ξ3)−M(x, ξ3)
≥ δΓx
log x+ o
(x
log x
)− log 2
x
log x+ o
(x
log x
).
Therefore, by Corollary 1.9
limx→∞
N(x)
x/ log x≥ lim
x→∞(δΓ − log 2 + o(1)) ≥ 1− log 2
which is the wanted claim.2
The estimate in (8) is the real obstacle to achieve an asymptotic formula for N(x).
Such estimate is connected with the range of validity of the Chebotarev Density
Theorem of the field K = Q(ζl, 21/l). As we have seen in Chapter 2, such a range
depends on the determination of zero-free regions for L(s, χl) and thus of the Dedekind
zeta function ZK(s). Indeed we have
46
Theorem 3.2 Assuming in the strong Hypothesis that for any prime l and for any
non-trivial zero β + iγ of the Dedekind zeta function ZK(s), β < 1 − clog T
, then for
almost all primes p, F∗p is generated by the first [2 log p] primes.
Proof. Using exactly the same notation of Theorem 3.1, we now choose ξ1 =
1/4√
log log x, ξ2 = log2 x and ξ3 = (log2 x)(log log x)2.
The estimate of the main and last terms in (2) is the same, for the third term,
again we use the Siegel-Walfisz Theorem (4) and Merten’s formula (6),
M(x, ξ2, ξ3) ≤∑
ξ2<l<ξ3
(1
l − 1li(x) + O
(x exp
(−A
√log x
)))
≤ x
log xlog
(2 log log x+ log log log x
2 log log x
)+ o
(x
log x
)
= o
(x
log x
).
Finally for the second term we use Theorem 2.8, and gather that
M(x, ξ1, ξ2, ) ≤∑
ξ1<l<ξ2
(1
l(l − 1)li(x) + O
(xl exp
(−A
√log x
)))
= o
(x
log x
),
and this concludes the proof.2
Remark: It can be proven that the minimal assumption necessary to prove that
N(x) ∼ π(x) in Theorem 3.1 is that the Dedekind zeta function ZK(s) has a zero-free
region of the type
σ > 1− c
l1/2 log d1/lK T
, T ≥ 0,
for any prime l large enough and for some absolute positive constant c.
Indeed this would yield to a version of Chebotarev Density Theorem for K valid
up to l < (log x)2, and the rest of the proof would work as in Theorem 3.2.
47
Finally note that using this approach improvements, in terms of determining a
significantly smaller set of generators of F∗p for almost all p, are not possible. The
choice r = log p is in fact imposed by the statement of Lemma 1.4. The next Section
is devoted to analyzing this aspect in a more detailed manner.
3.2 Relaxation of the Hypothesis and Improvements
Our first intention is to prove a version of Theorem 3.1 in which the number of
generators is optimal with respect to the method used. As we have already remarked,
the choice of the minimal number of generators of F∗p for a positive proportion of p’s
is imposed by Lemma 1.4. Precisely
Theorem 3.3 a) Let f be a (monotone) function of p with f(p) → +∞ for p→∞
and let Γp be the subgroup of F∗p generated by the classes of the first
f(p)log p
log log p
primes, then for a set of primes of density greater than 1− log 2, we have Γp = F∗p.
b) Let α be a real number with 0 < α < e − 2 and let Γ′p be the subgroup of F∗p
generated by the classes of the first
log p
α log log p
primes, then for a set of primes of density greater than 1− log(2 +α), it follows that
Γ′p = F∗p.
Proof: a) The proof starts in the same way as in Theorem 3.1, where we assumed
p ≥ x1/2 and noticed that, for every x,
Γp ⊇ Γr,p =< p1, . . . , pr >, with r =
[f(x)
2
log x
log log x
]
48
and
#p ≤ x | Γp = F∗p ≥ N(x, ξ1)−M(x, ξ1, ξ2)−M(x, ξ2, ξ3)−M(x, ξ3) (10)
where the notations, ξ1 and ξ2 are the same as in Theorem 3.1 and ξ3 will be chosen
later.
The estimate of the main term and the second term are exactly the same, while
this time the estimate of the last term of (10) using Lemma 1.4 is the following:
M(x, ξ3) ≤ #
p ≤ x | |Γr,p| ≤
x
ξ3
x
ξ3
(x
ξ3
)1/r
r log r
x
ξ3(log x)2/f(x)f(x) log x x(log x)1+ε(x)
ξ3(11)
where we have put ε = ε(x) = 2/f(x) + (log f)/ log log x and assumed that f(x)
log log x, say.
If we now choose ξ3 = (log x)2+ε log log x, we get that (11) is
x
log x log log x.
Finally we deal with the third term similarly as we did in Theorem 3.1, using the
Siegel-Walfisz Theorem and the Merten’s Formula:
M(x, ξ2, ξ3) ≤∑
ξ2<l<ξ3
(1
l − 1li(x) + O(x exp(−A
√log x))
)
≤ li(x)
(log
(log ξ3log ξ2
)+ O((log ξ2)
−1)
)+ O
(ξ3x exp−c
√log x
)
≤ x
log x
(log 2 + log
(1 + ε(x)/2
1− log(B(log log x)2)/ log log x
))+ o
(x
log x
)
=x
log xlog 2 + o
(x
log x
),
and this concludes the proof of a).
49
b) In this case we need to be a little more careful with the definition of r. We
can assume that p ≥ x1−ε(x) where ε(x) is a given function which is o( log log xlog x
), and
therefore we define
r =
[(1− ε(x))
log x
α log log x
]and the rest is as in a).
The last term is
M(x, ξ3) ≤ #
p ≤ x | |Γ′r,p| ≤
x
ξ3
x
ξ3
(x
ξ3
)1/r
r log r x
ξ3(log x)1+α/(1−ε(x)).
and choosing ξ3 = (log x)2+α/(1−ε(x)) log log x, we would make it o(x/ log x). Finally
the estimate for the third term is:
M(x, ξ2, ξ3) ≤ li(x)
(log
(log ξ3log ξ2
)+ O((log ξ2)
−1)
)+ O
(ξ3x exp−c
√log x
)
≤ x
log xlog
(2 +
α
1− ε(x)
)+ o
(x
log x
)
=x
log xlog(2 + α) + o
(x
log x
)
and this completes the proof.2
Now we turn our attention to another aspect. Note that neither the proof of
Theorem 3.1 nor the one of Theorem 3.3 use in any way the fact that each Γp is
generated by the first [log p] primes except for the fact that the sum of their logarithms
is log p log log p and that limr→∞ δΓ = 1. The statement remains true if we consider
a sequence a1, a2 . . . of multiplicatively independent integers such that for any r,
r∑i=1
log a1 r log r and limr→∞
δΓ = 1.
It is conceivable to ask if a choice of a1, . . . , ar exists for which we could prove a
stronger Theorem. That would amount to having a better estimate for the sum of
50
the logarithms. For this purpose one could set
Υ(r) = min
r∑
i=0
log ai | a1, . . . , ar, multiplicatively independent r−tuple
.
The following holds:
Proposition 3.4
Υ(r) = r log r +O(r).
Proof: For any multiplicatively independent a1, . . . , ar, we can assume
a1 ≥ 1, . . . , ar ≥ r, therefore
r∑i=1
log ai ≥r∑
i=1
log i = log r! = r log r − r + O(log r)
the last identity, by the Stirling formula. Thus
Υ(r) ≥ r log r + O(r).
The choice a1 = 2, . . . , ar = pr, the rth prime, and the Prime Number Theorem,
prove that
Υ(r) ≤r∑
i=0
log pr = pr + O(pr exp−A√
log pr) = r log r + O(r exp−A√
log r).2
Although many of the results that we will state can be extended to any sequence of
multiplicatively independent integers, from now on we will only consider the sequence
of the prime numbers.
It is now clear that the problem amounts to estimating the number of those primes
p ≤ x for which [F∗p : Γr] has a prime divisor in the range (log1−ε1(x) x, log2+ε2(x) x)
where εi(x) = o(1). We note that it is enough to restrict our attention to those
primes for which the prime divisor is in (log x, log2 x), since by the argument we
already used more than once (the Siegel-Walfisz Theorem and the Mertens Formula),
51
that the number of primes p for which p− 1 has a prime divisor in (log1−ε1(x) x, log x)
or in (log2 x, log2+ε2(x) x) is o(π(x)).
We can also assume that [F∗p : Γr] has exactly one prime divisor in (log x, log2 x).
Indeed, if p is a prime in the set under consideration for which this is not the case,
we would have
∃l1, l2, l1l2∣∣∣[F∗
p : Γr]∣∣∣ =⇒ |Γp| ≤
p− 1
l1l2≤ x
log2 x
and an application of Lemma 1.4 shows that the number of such primes is o(π(x)).
Finally for the same reason we can assume no divisors of [F∗p : Γr] are > log2 x.
Putting these remarks together, we have the following
Proposition 3.5 With the same notation of Theorem 3.3, for almost all primes p
up to x either
F∗p = Γr
or the index [F∗p : Γr] has exactly one prime divisor in the range (log x, log2 x) and no
divisor > log2 x, i.e.
N(x) =x
log x− A(x) + o
(x
log x
),
where
A(x) =p ≤ x | ∃!l ∈
(log x, log2 x
), l|[F∗
p : Γr], and [F∗p : Γr] ≤ log2 x
.2
This fact will be used in Section 3.3. We conclude the Section mentioning how
this argument can be extended to the case of any r →∞, in particular the following
holds:
Theorem 3.6 Let r be a function of p that tends to ∞, then for almost all primes
p, either Γr = F∗p or
∃!l, l|[F∗p : Γr] with l ∈
(log p, rp1/r log p
), and [F∗
p : Γr] < rp1/r log p.2
52
Corollary 3.7 Let α > 0 be fixed. For almost all primes p either F∗p is generated by
the classes of the first 1α
log plog log p
primes, or
∃!l, l|[F∗p : Γp] with l ∈
(log p, log2+α p
), and [F∗
p : Γp] < log2+α p.2
As we have seen in Proposition 3.5, the problem is now to determine upper bounds
for the quantity:
A(x) =p ≤ x | ∃!l ∈
(log x, log2 x
), l|[F∗
p : Γr], and [F∗p : Γr] ≤ log2 x
where we can suppose r log x. We already noticed that the Siegel-Walfisz Theorem
and the Merten’s Formula, give:
A(x) ≤ log 2x
log x+ o
(x
log x
).
The idea that allows one to improve this upper bound is coming from the Brun’s
Sieve, more precisely we will use the following result:
Lemma 3.8 Let Bn(x) be the number of primes up to x for which p−1 is not divisible
by any of the primes in the interval (log x, logn x), then we have
Bn(x) ∼ 1
nπ(x).
Proof: It is an application of the version of the Brun’s Sieve that is on Theorem
2.5′ at page 83 of [18] to the set:
A = p− 1 | p is primes, p ≤ x.
Hypothesis (R0) and (R1(k, a)) are easily satisfied, the latter using the Bombieri-
Vinogradov
Theorem.2
The application to our problem with the following:
53
Corollary 3.9 With the same notation as above, for any r,
A(x) ≤ 1
2
x
log x+ o
(x
log x
).
Proof: We have that
A(x) ≤p ≤ x | ∃l ∈
(log x, log2 x
), l|[F∗
p : Γr]
= π(x)−B2(x) ∼1
2π(x).2
We conclude with
Theorem 3.10 a) Let f be a (monotone) function of p with f(p) → +∞ for p→∞
and let Γp be the subgroup of F∗p generated by the classes of the first
f(p)log p
log log p
primes, then for at least half of the primes p, we have Γp = F∗p.
b) Let α be a real number with 0 < α and let Γ′p be the subgroup of F∗p generated
by the classes of the firstlog p
α log log p
primes, then for a set of primes of density greater than 12+α
, it results Γ′p = F∗p.2
Proof of b): The proof goes as the one in Theorem 3.3, except that to estimate
the third term we make use of Lemma 3.8 with n = 2 + α.2
54
3.3 A Density One Result
We will discuss in this section a method to find an estimate for the size of the set
Hm,r(x) =p ≤ x | |Γr| =
p− 1
m
where m is a given integer greater than one and r a function of p that tends to infinity.
The idea is that Γr is a subgroup of the cyclic group F∗p and therefore is itself
cyclic. For any integer m, m ≡ 1(modp), the subgroup of m-th powers is a subgroup
of F∗p of order (p−1)/m and since a finite cyclic group has a unique subgroup for every
divisor of its order, we deduce that Γr is the group of m-th powers mod p. Since a
group is made out of m-th powers if and only if it is generated by m-th powers, this
implies:
Hm,r(x) = p ≤ x | p ≡ 1(modm) and pi is an m-th power (modp) ∀i = 1, . . . , r .
If nm(p) is the least prime which is not congruent to an m-th power (modp), then
we can also write:
Hm,r(x) = p ≤ x | p ≡ 1(modm) and nm(p) > pr .
As r grows, the possibility that all the pi’s are m-th powers becomes less probable.
The idea is to find the minimum r such that Hm,r(x) is o( 1mπ(x)). We will do this
making use of the large sieve, the proof of which can be found in [6] or [2], that is:
Lemma 3.11 (The Large Sieve)
Let N be a set of integers contained in the interval 1, . . . , z and for any prime
p ≤ x, let Ωp = h(modp) | ∀n ∈ N , n 6≡ h(modp) and
L =∑q≤x
µ2(q)∏p|q
|Ωp|p− |Ωp|
,
55
then
|N | ≤ z + 3x2
L.2
In our case, let N = n ≤ z | ∀q|n, q < pr and note that if p ∈ Hm,r(x), then
Ωp ⊃ h(modp) | h is not an m-th power (modp)
therefore, for such p’s, |Ωp| ≥ p− 1− (p− 1)/m and
L ≥∑
p∈Hm,r(x)
|Ωp|p− |Ωp|
≥ m− 1
2|Hm,r(x)|.
Applying the Large Sieve with z = x2, we get:
Theorem 3.12 Let Ψ(s, t) = #n ≤ s‖ ∀q|n, q < t, then
Hm,r(x) ≤8x2
(m− 1)Ψ(x2, pr).2
Estimating the function Ψ(z, y) is a classical problem in Number Theory. In 1985,
D. Hensley proved (see [22]) the following:
Lemma 3.13 Let u = log zlog y
and let ρ(u) be the function determined by:
ρ(u) = 1 if 0 ≤ u ≤ 1;
uρ′(u) = −ρ(u− 1) if u > 1,
then, for 1 + log log z ≤ log y ≤ (log log z)2, and ε > 0,
Ψ(z, y) zρ(u) exp(−u exp(−(log y)(3/5−ε))).2
In our case, this gives:
56
Corollary 3.14 Let r be a function of x such that log pr ∈ [1+log log x2, (log log x2)2]
and let u = 2 log x/ log pr, then
Hm,r(x) m−1 1
ρ(u)exp
(u exp
(−u exp
(− log(3/5−ε) pr
))).2
An asymptotic formula for ρ(u) was found by de Bruijn in [10] and is the following:
Lemma 3.15 Let u > 0, then
ρ(u) = exp
−u
(log u+ log log u− 1 +
(log log u)− 1
log u+ O
((log log u)2
log2 u
)).2
In our case we get the following:
Theorem 3.16 If pr ≥ log2 x then
Hm,r(x) 1
m
x
exp
log x2 log log x
= o(π(x)).
Proof: From Corollary 3.14 and the asymptotic formula of Lemma 3.15, we can
write the estimate:
Hm,r(x) 1
mexp(u(log u+ log2 u− 1 + O
(log log u
log u
))).
where u = 2 log xlog pr
. Now, take pr ≥ log2 x, and note that
u = log xlog2 x
; log u = log2 x− log3 x; log2 u = log3 x+ log(1− log3 x
log2 x
)and log u log2 x; log2 u ≤ log3 x−
log3 xlog2 x
log3 x.
Therefore
mHm,r(x) exp
log x
log2 x
(log2 x− log3 x+ (log3 x−
log3 x
log2 x)− 1 + O
(log3 x
log2 x
))
exp
log x
(1− 1
log2 x+ O
(log3 x
log22 x
)) x
exp
log x2 log log x
= o(π(x)).2
57
Remark:
The choice of pr = log2 x is not optimal in Theorem 3.16. A simple but long calcu-
lation shows that if pr =(
log xe
)2, then the asymptotic formula on Lemma 3.15 gives
the estimate1
ρ(u) x
(1+O
(log2
3 x
log32
x
)),
which is useless to our purpose. However, if we fix δ < 1 and set pr =(
log xeδ
)2, then
the same calculation gives
1
ρ(u) x
(1− 1−δ
log2 x− 1−δ2
log22
x+O(
log23 x
log32
x
)),
which is a valid estimate and is the optimal one.
We are now ready to prove:
Theorem 3.17 Let Γr = 〈p1, . . . , pr〉 be the subgroup of F∗p generated by all the
primes up to pr = log2 p (r ∼ log2 plog log p
), then for almost all primes p,
Γr = F∗p
Proof: We want to estimate the size of the set
S =p ≤ x |
[F∗
p : Γr
]> 1
.
The index[F∗
p : Γr
]can be at most x since it is a divisor of p− 1.
Since we may suppose p > x1−ε, we can take pr = log2 p > (1 − ε)2 log2 x and
apply Theorem 3.16, to S,
S =x∑
m=1
Hm,r(x) =
(x∑
m=1
1
m
)x
exp
log x2 log log x
= o(π(x)).2
Remarks:
58
a) This is an improvement with respect to the result of Burgess and Elliot of [5]
deduced in Proposition 1.11 where for almost all primes ≤ x, the size of pr (least
primitive root) was proven to be ≥ log2 x(log log x)4;
b) Improvements to this result using this approach do not stay in the possibility to
apply a stronger version of Lemma 3.13 (which exists in the literature, see for example
the work of A. Hildebrand in [23] or Canfield, Erdos and Pomerance in [8]). It is the
asymptotic formula of De Bruijn for the function ρ(u) that forces a choice pr of size
close to log2 x.
59
4 MORE ON PRIMITIVE ROOTS
This Chapter is devoted to the problem of finding a uniform asymptotic formula for
the number of primes p up to x such that s distinct numbers (which for simplicity
we suppose to be prime) are all at the same time primitive roots (modp). It turns
out that, under the assumption of the G.R.H., there is always a positive density of
primes with such a property.
Heuristically, the probability that a prime l divides one the indexes [F∗p : 〈p1〉]
or [F∗p : 〈p2〉] is the probability that p splits completely in the fields Q(ζl, p
1/l1 )
and Q(ζl, p1/l2 ), minus the probability that p splits completely in the compositum
Q(ζl, p1/l1 , p
1/l2 ). By the Chebotarev density Theorem, we get that the probability
that l does not divide both the indexes is
1−(
1
[Q(ζl, p1/l1 ) : Q]
+1
[Q(ζl, p1/l2 ) : Q]
− 1
[Q(ζl, p1/l1 , p
1/l2 ) : Q]
).
Multiplying for all primes l, we get the formula:
δ =∏
l prime
(1− 2l − 1
l2(l − 1)
).
A natural generalization of this argument to the case of r distinct primes with the
application of an inclusion exclusion principle, yields to:
δ =∏
l prime
(1− 1
l − 1
[1−
(1− 1
l
)s]).
We will prove that this heuristic argument is correct with the assumption of the
Generalized Riemann Hypothesis and some adjustments of the same type of those
noticed by Lehmer in the case of the Artin Conjecture for primitive roots. This will
be applied in Section 4.3 to the problem of determining the least prime primitive root
modp for almost all primes p.
60
4.1 Another Generalization of Hooley’s Theorem
We present in this section a very natural generalization of Hooley’s Theorem for
primitive roots.
Theorem 4.1 Let P = p1, . . . , ps be a set of odd primes, L(d1, . . . , ds) be the
compositum field:
L(d1, . . . , ds) =s∏
i=1
Q(ζdi, p
1/di
i ),
nd1,...,ds = [L(d1, . . . , ds) : Q], and let
δP =∞∑
d1=1,...,ds=1
µ(d1) · · ·µ(ds)
nd1,...,ds
.
Define
NP(x) = #p ≤ x | ∀i = 1, . . . , s pi is a primitive root (mod p).
Then, if the Generalized Riemann Hypothesis holds for the fields L(d1, . . . , ds), we
have that
NP(x) = δPx
log x+ O
(x log log x
log2 xcs0
s∑i=1
log pi
)for some absolute constant c0, uniformly respect to s = |P| and p1, . . . , ps.
Remark: It is not straightforward to see that δP is well defined nor that it is non
zero. Indeed, the series defining the density, converges absolutely. We will assume it
for the moment and prove it in the next section in Corollary 4.12.
Proof: We will follow the same approach of Hooley who first noticed that in
order p1, . . . , ps be all primitive roots for the same prime p, one has to have that
∀l prime, l 6 |[F∗p : 〈pi〉],∀i = 1, . . . , s,
therefore
NP(x) = NP(x, y) +MP(x, y)
61
where
NP(x, y) = #p ≤ x | ∀l prime, l < y and ∀i = 1, . . . , s l 6 |[F∗
p : 〈pi〉]
and
MP(x, y) = #p ≤ x | ∃l prime, l > y with l|[F∗
p : 〈pi〉], for some i = 1, . . . , s.
We choose y = 16s
log x for a reason that will become clear later.
Step 1): MP(x, y) x log log x
log2 xs
s∑i=1
log pi.
Clearly
MP(x, y) ≤s∑
i=1
Mpi(x, y),
therefore it is enough to show that, uniformly with respect to P ,
Mq(x, y) x log log x
log2 xs log q.
This was proven already by Hooley in his original paper [26], and we will report it
here just for completeness.
Note that
Mq(x, y) ≤ |A|+ |B|+ |C|,
where
A =p ∈Mq(x, y)| ∃l|[F∗
p : 〈q〉], l > x1/2 log x
;
B =p ∈Mq(x, y)| ∃l|[F∗
p : 〈q〉], x1/2
log2 x< l ≤ x1/2 log x
;
C =p ∈Mq(x, y)| ∃l|[F∗
p : 〈q〉], y ≤ l ≤ x1/2
log2 x
.
|A| can be estimated as follows:
l|[F∗p : 〈q〉] implies that
qp−1
l ≡ 1 (mod p).
62
Therefore, since l > x1/2 log x and p ≤ x, any prime in A must divide the positive
product ∏m≤x1/2 log−1 x
(qm − 1).
Now note that the number of divisors of a natural number N is O(logN), therefore
|A| ∑
m≤x1/2 log−1 x
m log q x
log2 xlog q.
|B| can be estimated as follows:
Retaining only the condition l|p− 1 for the primes p ∈ B, we get
|B| ≤∑
x1/2
log2 x<l≤x1/2 log x
π(x, l, 1).
By the Brun-Titchmarsh Theorem, we know that
π(x, l, 1) x
(l − 1) log(x/l).
We therefore deduce that
|B| x
log x
∑x1/2
log2 x<l≤x1/2 log x
1
l x
log2 x
∑x1/2
log2 x<l≤x1/2 log x
log l
l,
from which it follows from the easier Merten’s formula that
|B| x
log2 x
(log
(x1/2 log x
)− log
(x1/2
log2 x
)+ O(1)
)= O
(x log log x
log2 x
).
|C| can be estimated as follows:
We have already noticed that l|[F∗p : 〈q〉] is equivalent to the statement that p splits
completely in the field Q(ζl, q1/l), the version of the Chebotarev Density Theorem
that assumes the validity of the Generalized Riemann Hypothesis for such fields is:
P (x, l) = #p ≤ x | p splits completely in Q(ζl, q1/l) =
63
1
l(l − 1)li(x) + O(x1/2 log x · l · q).
If we use this formula we get:
|C| =∑
y≤l≤ x1/2
log2 x
P (x, l) =∑
y≤l≤ x1/2
log2 x
[1
l(l − 1)li(x) + O(x1/2 log x · l · q)
]=
O
(1
y
x
log x
)+ O
(x
log2 xlog q
).
Taking into account that y = 16s
log x, we get
|C| = O
(x
log2 xs log q
),
which is the desired estimate.
Step 2): We can now turn our attention to NP(x, y). We claim that
NP(x, y) =∗∑a1
· · ·∗∑as
µ(a1) · · ·µ(as)P (x, a1, . . . , as) (1)
where the ∗ over the sums means that the sums are extended to those values of ai for
which all its prime divisors are less than y (note that this implies ai < e2y = x1/3s),
and
P (x, a1, . . . , as) = #p ≤ x | ai
∣∣∣[F∗p : 〈pi〉] ∀i = 1, . . . , s
.
This claim can be proven by induction on s: If s = 1 then we have the standard
inclusion-exclusion principle:
Np1(x, y) = π(x)−∑l<y
P (x, l) +∑
l<y,l′<y
P (x, ll′)− · · · =∗∑
a1=1
µ(a1)P (x, a1).
Similarly, for 0 ≤ t ≤ s, define
Pt(x, at+1, . . . , as) = #p ∈ Np1,...,pt(x, y)∣∣∣ai
∣∣∣[F∗p : 〈pi〉] ∀i = t+ 1, . . . , s.
Clearly
P0(x, a1, . . . , as) = P (x, a1, . . . , as)
64
and the recursive relation holds;
Pt(x, at+1, . . . , as) =∗∑at
µ(at)Pt−1(x, at, . . . , as).
Hence inductively
NP(x, y) = Ps(x) =∗∑as
µ(as) . . .∗∑a1
µ(a1)P0(x, a1, . . . , as).
Now note that the condition ai
∣∣∣[F∗p : 〈pi〉] is equivalent to
p splits completely in Q(ζai, p
1/ai
i ),
and that a prime splits completely in a set of fields if and only if it splits completely
in their compositum.
Therefore ai
∣∣∣[F∗p : 〈pi〉] for all i = 1, . . . , s, if and only if p splits completely in
s∏i=1
Q(ζai, p
1/ai
i ) = Q(ζ[a1,...,as], p1/a1
1 , . . . , p1/ass ).
We gather that
P (x, a1, . . . , as) = # p ≤ x | p splits completely in L(a1, . . . , as)
and the Generalized Riemann Hypothesis allows us to write the Chebotarev Density
formula:
P (x, a1, . . . , as) =1
na1,...,as
li(x) + O
(x1/2
(log x+
logDa1,...,as
na1,...,as
)).
where Da1,...,as is the discriminant of L(a1, . . . , as). Recall that by the Hensel Inequal-
ity (See page 259 of [42]) we can write
logDa1,...,as
na1,...,as
≤ log[a1, . . . , as] +s∑
i=1
log pi,
65
since only p1, . . . , ps and the primes dividing [a1, . . . , as] ramify in L(a1, . . . , as). Now
substitute inside (1) and get:
NP(x, y) =∗∑a1
· · ·∗∑as
µ(a1) · · ·µ(as)
li(x)
na1,...,as
+ O
(x1/2
(log x+
logDa1,...,as
na1,...,as
))=
δP li(x) + O
∑(a1,...,as)∈S
µ2(a1) · · ·µ2(as)
na1,...,as
li(x)+
O
∑a1<e2y ,...,as<e2y
x1/2
(log x+
s∑i=1
log p1 + log[a1, . . . , as]
) (2)
where S is the set of s-tuples of positive integers where at least one of the component
is greater that y. We will prove later in Proposition 4.4 that
∑(a1,...,as)∈S
µ2(a1) · · ·µ2(as)
na1,...,as
cs1y,
for some absolute constant c1 . In our case 1/y s/ log x therefore (2) is equal to
δPx
log x+ O
(x
log2 xscs1
s∑i=1
log pi
)+ O
x1/2∑
a1<e2y ...as<e2y
log[a1, . . . , as]
.Finally note that if a1, . . . , as are square-free numbers with prime divisors less than
y, then also [a1, . . . , as] has the same property, thus
∑a1<e2y ...as<e2y
log[a1, . . . , as] ∑
a1<e2y ...as<e2y
y ye2sy x1/3 log x
Hence
NP(x, y) = δPx
log x+ O
(x
log2 xcs0
s∑i=1
log pi
).
which, together with step 1) proves the Theorem.2
66
4.2 Calculation of the Densities
We can now give the expression for the density δP , and as we did in Section 1.2, the
first step is to calculate the dimension of the fields
L = La1,...,as =s∏
i=1
Q(ζai, p1/ai
)= Q(ζ[a1,...,as], p
1/a1
1 , . . . , p1/ass ),
for any s-tuple a1, . . . , as of square-free integers. This is done in the following:
Theorem 4.2 Let n = na1,...,as = [L : Q] ,M = [a1, . . . , as] and suppose P is the
product of those pi such that pi|M and ai is even. Let t be the number of prime
factors of P , then
n =φ(M)a1 · · · as
2α,
with
α =
t if ∀q|P, q ≡ 1 (mod 4),
t− 1 if ∃q|P,with q ≡ 3 (mod 4).
Proof: The argument is similar to the one in the proof of Theorem 1.5. We can
suppose, with out loss of generality, that P = p1 · · · pt, and define
C0 = Q(ζM), Ci = Ci−1(p1/ai
i ).
Clearly L = Cs and
[L : Q] = [Cs : Cs−1] · · · [C2 : C1]φ(M).
Step 1): For 1 ≤ i ≤ s, it results [Ci : Ci−1] = ai or ai/2 .
Since xa1 − pi splits into linear factors over Ci−1, we have that [Ci : Ci−1] =ai
d, if q|d
is a prime, then we have the fields:
Ci−1 ⊆ Ci−1(p1/qi ) ⊆ Ci.
67
Now, either q = 1 or
[Ci−1(p1/qi ) : Ci−1] |[Ci : Ci−1] =
ai
d
and since ai is square-free, we deduce that p1q
i ∈ Ci−1. Either pi ∈ C0 which imply
q = 2 since the only subfields of a cyclotomic field of the type Q(p1/qi ) are quadratic,
or C0(p1/qi ) is a Kummer extension of degree q of C0. Now by Galois Theory, we get
that such an extension has to be of the following type:
C0(p1/qi ) = C0
((ps1 · · · psk
)1/q),
where 1 ≤ s1 ≤ s2 · · · ≤ sk ≤ i − 1 Finally, the Theory of Kummer extensions, (See
Lemma 3 at page 160 of Cassels and Frohlich [7]) implies that there exists 0 ≤ i ≤ q−1
such that (pi
(ps1 · · · psk)i
)1/q
∈ C0,
which again implies q = 2.
Step 2): [Ci : Ci−1] = ai for t < i ≤ s.
In the case ai odd then clearly Step 1) implies Step 2), thus suppose ai is even and
[Ci : Ci−1] = ai/2. In this case, we have that√pi is in Ci−1 because ai is square-
free ([Ci−1(√pi) : Ci−1] |ai/2). This implies that pi ramifies in Ci−1, but since, by
the Kummer Theory, the only primes that ramify in Ci−1 are p1, . . . , pi−1 and those
dividing M , we get a contradiction and conclude that [Ci : Ci−1] = ai.
This also implies that [Cs : Ct] = at+1 · · · as.
Step 3): If every prime dividing P is ≡ 1( mod 4), then [Ci : Ci−1] = ai/2 for every
1 ≤ i ≤ t.
From the Theory of Cyclotomic fields we know that a generic quadratic subfield of
C0 = Q(ζM) is of the following type:
Q
√(−1
D
)D
, where D |M ,
68
since(−1q
)= 1 if and only if q ≡ 1(mod4), we deduce that
√qi ∈ C0 and the Galois
group of Ci over Ci−1 is generated by the map
σ : p1/ai
i 7→ ζ2aip
1/ai
i
(note that σ(√pi) = (σ(p
1/ai
i ))ai/2 =√pi), which has clearly order ai/2.
In this case we have [Ct : C0] = a1 · · · at/2t.
Step 4): If it exists q |P with q ≡ 3( mod 4) (we assume, without loss of generality,
that q = p1), then [C1 : C0] = a1 and [Ci : Ci−1] = ai/2 for every 1 < i ≤ t.
The assumption [C1 : C0] = a1/2 would imply again that√p1 ∈ C0. By the same
argument of Step 3), this implies that the Legendre symbol(−1p1
)= 1; which is a
contradiction. Therefore we are left to show the second part of the statement of this
Step.
If 1 < i ≤ t then√pi ∈ C1 because, either pi ≡ 1(mod4) and thus
√pi ∈ C0 ⊂ C1,
or pi ≡ 3(mod4) and√p1pi ∈ C0 hence
√pi =
√p1pi/
√p1 ∈ C1. In both cases, the
Galois group of Ci over Ci−1 is generated by the map
σ : p1/ai
i 7→ ζ2aip
1/ai
i
which again has order ai/2.
Finally, in this case we have [Ct : C0] = a1 · · · at/2t−1 and this concludes the proof.2
Corollary 4.3 We have the following lower bound for the dimension na1,...,as of the
field La1,...,as over Q:
na1,...,as ≥φ([a1, . . . , as])a1 · · · as
2s.2
We have now enough tools to prove the property we used during the proof of
Theorem 4.1.
69
Proposition 4.4 Recall that S = (a1, . . . , as) ∈ Ns | ∃i with ai > y. It results
that ∑(a1,...,as)∈S
µ2(a1) · · ·µ2(as)
na1,...,as
cs1y
(3)
for some absolute constant c1.
Before proving Proposition 4.4, we need the following technical Lemma:
Lemma 4.5 Consider the multiplicative function dt(n) defined as the number of ways
to write n as product of t natural numbers, and denote:
σ(t) =∏
l prime
1 +1
l
∑k≥1
dt(lk)
lk
.we have:
σ(t) ≤ ζ(2)2t.
Proof: Note that dt(l2k−1) ≤ dt(l
2k), therefore
1 +1
l
(dt(l)
l+dt(l
2)
l2+ · · ·
)≤ 1 + 2
(dt(l
2)
l2+dt(l
4)
l4+ · · ·
)
≤(
1 +dt(l
2)
l2+dt(l
4)
l4+ · · ·
)2
=(1 +
1
l2+
1
l4+ · · ·
)2t
.
Hence
σ(t) ≤∏
l prime
(1 +
1
l2+
1
l4+ · · ·
)2t
= ζ(2)2t.2
Proof of Proposition 4.4: First note that from Corollary 4.3, we get that
na1,...,as ≥φ([a1, . . . , as])a1 · · · as
2s,
therefore (3) is
2s∑S
µ2(a1) · · ·µ2(as)
a1 · · · asφ([a1, . . . , as]), (4)
70
But the sum on (4) is completely symmetric on the ai’s, therefore is
2ss∑a1>y
(a2,...,as)∈Ns−1
µ2(a1) · · ·µ2(as)
a1 · · · asφ([a1, . . . , as])
2ss∑
a1>y
µ2(a)
a
∑(a2,...,as)∈Ns−1
1
a2 · · · asφ([a2, . . . , as]). (5)
Using the multiplicative function defined above and the function γ defined as
γ(n) =∏p|np,
we can write that (5) is equal to:
2ss∑a>y
µ2(a)
a
∞∑b=1
ds−1(b)
φ(γ(ab))= 2ss
∑a>y
µ2(a)
φ(a)a
∞∑b=1
ds−1(b)φ((a, b))
φ(γ(b)).
Since all functions inside the second sum are multiplicative, we can write (5) as
2ss∑a>y
µ2(a)
φ(a)a
∏l prime
1 +
φ((a, l))
φ(l)
(ds−1(l)
l+ds−1(l
2)
l2+ · · ·
)
2ssσ(s− 1)∑a>y
µ2(a)
φ(a)a
∏l|a
∑k≥0
ds−1(lk)
lk
.If we can prove the estimate:
∑a>y
f(a)
a2 σ(s− 1)
y(6)
where
f(a) =aµ2(a)
φ(a)
∏l|a
∑k≥0
ds−1(lk)
lk= µ2(a)
(a
φ(a)
)s
,
then the estimate for the function σ(t) of Lemma 4.5 would imply the claim.
From the Theory of Dirichlet series, we know that (6) is
∫ ∞
y
F (x)
x3dx,
71
where F (x) =∑
n≤x f(n) is the average of f . If we could prove the asymptotic
formula:
F (x) ∼ σ(s− 1)x, (7)
for x that tends to infinity, then we would have that (6) is
∫ ∞
y
σ(s− 1)x
x3dx =
σ(s− 1)
y.
To prove (7), we make use of the Dirichlet series:
H(z) =∞∑
n=1
f(n)
nz. (8)
Since f(n) (log log n)s, we know that H(z) converges in the semi-plane <(z) > 1
and we can write the Euler product expansion:
H(z) =∏l
1 +1
lzl
l − 1
∑k≥0
ds−1
lk
= ζ(z)K(z)
where
K(z) =∏l
1 +1
lzl
l − 1
∑k≥0
ds−1
lk
(1− 1
lz
)converges for <(z) > 0. This decomposition gives an analytic continuation for H(z)
and therefore we can calculate the residue at z = 1 of H(z) which is going to be
limz→1
(z − 1)ζ(z)K(z) = K(1) =∏l
1− 1
l+
1
l
1 +∑k≥1
ds−1(lk)
lk
= σ(s− 1).
The Ikehara Tauberian Theorem (see [33], page 311) implies the claim of (7) and the
Proposition results proven.2
Remark: The details used in the last part of the proof of Proposition 4.4 are
missing from the original proof of Hooley in the case s = 1. In that circumstance
the level of precision that we need here for the application on Section 4.3 was not
required.
72
If nd1,...,ds is the dimension over Q of∏s
i=1 Q(ζdi, p
1/di
i ) then we have seen that
whenever ([d1, . . . , ds], p1 · · · ps) = 1, it results
nd1,...,ds = d1 · · · dsφ([d1, . . . , ds]),
this leads us to consider the function:
∆s =∞∑
a1=1
· · ·∞∑
as=1
µ(a1) · · ·µ(as)
a1 · · · asφ([a1, . . . , as]).
Our goal is to show that ∆s 6= 0 and that lims→∞ ∆s = 0, the best way to do this
is again to express ∆s as an Euler product. This will also confirm the heuristic
argument illustrated at the beginning of this Chapter.
Proposition 4.6
∆s =∏
l prime
(1− 1
l − 1
[1−
(1− 1
l
)s]).
We need two lemmas:
Lemma 4.7 For any prime l, let it be given a function αl, and define
Γt+1(αl) =∞∑
a1=1
. . .∞∑
at+1=1
µ(a1) . . . µ(at+1)
a1 · · · at+1
∏l|[a1,...,at+1])
αl
, (9)
(note that Γs(1/(l − 1)) = ∆s) then:
Γt+1(αl) = Γt
(αll − 1
l − αl
) ∏q prime
(1− αq
q
).
Proof: The right hand side of (9) is equal to:
∞∑a1=1
· · ·∞∑
at=1
µ(a1) . . . µ(at)
a1 · · · at
∏l|[a1,...,at])
αl
∞∑x=1
µ(x)
x
∏l
∣∣∣ [a1,...,at,x]
[a1,...,at]
αl
, (10)
73
where we just renamed at+1 to be x. Note that for any multiplicative function f(n)
and H ∈ N, if we define
F (n) = µ(n)f([H,n]/H),
then F (n) is again multiplicative. If we take H = [a1, . . . , at] and f(n) =∏
l|n αl, we
get that (10) is equal to
∞∑a1=1
· · ·∞∑
at=1
µ(a1) · · ·µ(at)
a1 · · · at
∏l|[a1,...,at]
αl
∏q prime
1− 1
q
∏l
∣∣∣ [a1,...,at,q]
[a1,...,at]
αl
. (11)
Since l∣∣∣ [H,q]
Hif and only if l = q and q 6 |H, we gather that (11) is equal to:
∞∑a1=1
· · ·∞∑
at=1
µ(a1) · · ·µ(at)
a1 · · · at
∏l|[a1,...,at]
αl
∏l|[a1,...,at]
(1− 1
l
) ∏q prime
q 6|[a1,...,at]
(1− αq
q
).
Multiplying and dividing by the missing terms, we get the claim.2
Lemma 4.8 Define inductively the functions βi = βi(l) in the following :
β1 =1
l − 1and βn = βn−1
l − 1
l − βn−1
,
then:
∆s =∏
l prime
s∏n=1
(1− βn(l)
l
).
Proof: By induction on s, the case s = 1 being the definition of the Artin
constant. Lemma 4.7 implies that:
∆s = Γs(β1) = Γs−1(β2)∏
l prime
(1− β1(l)
l
)=
Γs−2(β3)∏
l prime
(1− β1(l)
l
)(1− β2(l)
l
)=
...
Γ1(βs)∏
l prime
[s−1∏i=1
(1− βi(l)
l
)].
74
Finally
Γ1(βs) =∞∑
a=1
µ(a)
a
∏l|aβs(l) =
∏l prime
(1− βs(l)
l
),
and the claim follows.2
First Proof of Proposition 4.6: Note that by the inductive definition of the
βi’s, we have that(1− βi(l)
l
)(1− βi+1(l)
l
)=
(1− βi(l)
(1−
(1− 1
l
)2))
and, more in general(1− βi(l)
l
)(1− βi+1(l)
(1−
(1− 1
l
)k))
=
(1− βi(l)
(1−
(1− 1
l
)k+1))
.
Finally
∆s =∏
l prime
[(1− β1(l)
l
)· · ·
(1− βs−1(l)
l
)(1− βs(l)
l
)]=
∏l prime
[(1− β1(l)
l
)· · ·
(1− βs−2(l)
l
)(1− βs−1(l)
(1−
(1− 1
l
)2))]
...∏l prime
[(1− β1(l)
l
)(1− β2(l)
(1−
(1− 1
l
)s−1))]
.
Substitute β1(l) = 1/(l − 1), and get the claim.2
Second Proof of Proposition 4.6: If we define
f(m) =∑
a1,...,as[a1,...,as]=m
µ(a1) · · ·µ(as)
a1 · · · as
,
then it results
∆s =∞∑
m=1
µ2(m)f(m)
φ(m)
75
since the lowest common multiple of square-free integers is itself square-free.
We now claim that f(m) is multiplicative, which imply:
∆s =∏
l prime
(1 +
f(l)
l − 1
). (12)
Indeed, if m and m′ are coprime integers, then the map
(a1, . . . , as), (b1, . . . , bs) 7−→ (a1b1, . . . , asbs)
is a bijection from the set of s-tuples of integers with lowest common multiple m cross
the set of r-tuples of integers with lowest common multiple m′ to the set of s-tuples
of integers with lowest common multiple mm′, whose inverse map is given by:
(c1, . . . , cs) 7−→ ((c1,m), . . . , (cs,m)), ((c1,m′), . . . , (cs,m
′)).
We gather that
f(m)f(m′) =∑
a1,...,as[a1,...,as]=m
∑b1,...,bs
[b1,...,bs]=m′
µ(a1b1) · · ·µ(asbs)
a1b1 · · · asbs=
∑c1,...,cs
[c1,...,cs]=mm′
µ(c1) · · ·µ(cs)
c1 · · · cs= f(mm′).
Finally, if [a1, . . . , as] = p, then each ai can be equal to 1 or to p, and each possibility
is possible except ai = 1∀i = 1, . . . , s. Hence
f(p) = s−1
p+
s
2
1
p2+ · · ·+
s
k
(−1)k
pk+ · · ·+ (−1)s
ps=
(1− 1
p
)s
− 1.
Substitute in (12) and get the claim.2
Corollary 4.9 With the same notation of Theorem 4.1, we have that:
∆s = O
(2−s 1
log s
).
76
Proof: For any fixed N > 0, we have that:
2s∆s ≤∏l<Nl6=2
(1− f(l)
l − 1
)
where f(l) =(1−
(1− 1
l
)s). Note that in such a range for l it results
f(l) ≥(1−
(1− 1
N
)s)∼ 1− e−
sn ,
as N tends to infinity. Hence
2s∆s ≤ exp
∑l<Nl6=2
log
(1− f(N)
l − 1
) exp−
∑l<Nl6=2
(1
l − 1
)f(N)
exp−
(log logN)(1− e−
sN ).
Now take N = s/ log s and get
2s∆s 1
log sexp
(log log(s/ log s)s−1
) 1
log s.2
Theorem 4.10 Let P = p1, . . . , ps be a set of odd primes, suppose P is the subset
of P of those primes congruent to 1(mod4). With the same notation of Theorem 4.2
and Proposition 4.6, it results
δP =∞∑
a1=1
· · ·∞∑
as=1
µ(a1) · · ·µ(as)
na1,...,as
=1
2∆s
∏p∈P
1
1 + αp
+∏p∈P
1 + αp −(−1p
)αp
1 + αp
,where αp = 1
p−1
((1− 1
p
)s− 1
).
Proof: To make the notation lighter, we will indicate the s-tuple (a1, . . . , as) by
a, the product a1 · · · as by a and µ(a1) · · ·µ(as) by µ(a). We also say that a is odd if
all its components are odd.
77
Furthermore, for any subset I of [s]def=1, . . . s, we denote by PI the product of
elements in I and by I the subset of I of those i’s for which pi ≡ 1(mod4). Now call
[a]I the set of s-tuples of integers for which ai is even for all i ∈ I and ai is odd for
all i 6∈ I. It is clear that [a]II∈[s] is a partition of Ns, therefore
δP =∑a
µ(a)
na
=∑I⊆[s]
∑a∈[a]I
µ(a)
na
=∑I⊆[s]
(−1)|I|
2|I|∑
a odd
2βIµ(a)
aφ([a])
where, if Q = (PI , [a]),
βI =
ν(Q) if Q |PI
ν(Q)− 1 otherwise
the possibility Q = 1 belonging to the first case. We gather that
∑a odd
2βµ(a)
aφ([a])=∑J⊆I
∑a odd
PJ=(PI ,[a])
2βµ(a)
aφ([a])=∑J⊆I
2ν(PJ )∑
a oddPJ=(PI ,[a])
µ(a)
aφ([a])+∑J 6⊆IJ⊆I
2ν(PJ )−1∑
a oddPJ=(PI ,[a])
µ(a)
aφ([a])
=1
2
∑J⊆I
2ν(PJ )∑
a oddPJ=(PI ,[a])
µ(a)
aφ([a])+∑J⊆I
2ν(PJ )∑
a oddPJ=(PI ,[a])
µ(a)
aφ([a])
.Now note that for J ⊆ I, the condition PJ = (PI , [a]) is equivalent to PJ |[a] and
([a], PI−J) = 1. As we did during the second proof of Proposition 4.6, we can write
∑a odd
PJ=(PI ,[a])
µ(a)
aφ([a])=
∞∑m=1
m odd,PJ |m,(m,PI−J )=1
µ2(m)
φ(m)
∑a odd[a]=m
µ(a)
a, (13)
again the function inside is multiplicative, thus we can write that (13) is equal to
∞∑m=1
m odd,PJ |m,(m,PI−J )=1
µ2(m)
φ(m)
∏l|m
[(1− 1
l
)s
− 1]
=1
φ(PJ)
∏l|PJ
[(1− 1
l
)s
− 1] ∞∑
m=1m odd, (PI ,m)=1
µ2(m)
φ(m)
∏l|m
[(1− 1
l
)s
− 1]
=
2s∆s
∏p|PJ
αp
∏p|PI
1
1 + αp
.
78
Putting everything together,
δP = 2s∆s
∑I⊆[s]
(−1)|I|
2|I|+1
∑J⊆I
2J∏p|PJ
αp
∏p|PI
1
1 + αp
+∑J⊆I
2J∏p|PJ
αp
∏p|PI
1
1 + αp
.Finally
∑I⊆[s]
(−1)|I|
2|I|∑J⊆I
2J∏p|PJ
αp
∏p|PI
1
1 + αp
=∑I⊆[s]
(−1)|I|
2|I|∏p|PI
1 + 2αp
1 + αp
=1
2s
s∏i=1
1
1 + αp1
and similarly
∑I⊆[s]
(−1)|I|
2|I|∑J⊆I
2J∏p|PJ
αp
∏p|PI
1
1 + αp
=∑I⊆[s]
(−1)|I|
2|I|∏p|PI
(1 + 2αp)∏p|PI
1
1 + αp
=
1
2s
s∏i=1
p1≡1 mod 4
1
1 + αp
s∏i=1
p1≡3 mod 4
(2− 1
1 + αp
)=
s∏i=1
(1−
(−1
p
)αp
1 + αp
)
therefore the claim.2
We conclude this Section with two Corollaries:
Corollary 4.11 Under tha same assumptions of Theorem 4.1, if every prime in P
is congruent to 1(mod4) then the density of primes for which all elements of P are
primitive roots, is ∏l 6∈P
(1 + αl) .2
Corollary 4.12 For any set of odd primes P, δP is a well defined number.2
79
4.3 An Application to the Least Prime Primitive Root
In this last section we apply Theorem 4.1 to the classical problem of the study of
the function G(p) defined as the least prime primitive root (modp). More precisely,
by the use of the inclusion exclusion principle, we determine a uniform asymptotic
formula for the number of primes p ≤ x such that G(p) < r.
Theorem 4.13 With the same notation and hypothesis of Theorem 4.1, let qn be the
n-th odd prime,
Tr(x) = #p ≤ x | ∃i ≤ r | qi is a prime primitive root (mod p), ,
and
δr =∑
P⊆p1,...,pr(−1)|P|δP
(δ∅ = 1). We have that
Tr(x) = (1− δr)x
log x+ O
(x log log x
log2 xCr
2
)
for some absolute constant C2, uniformly respect to r.
Proof: Let Sr be the number of primes p up to x such that none of the first r
primes is a primitive root modp.
As a straightforward application of the inclusion exclusion principle, we get that
Sr =∑
P⊆p1,...,pr(−1)|P|NP(x)
where as in Theorem 4.1,
NP = # p ≤ x |∀q ∈ P , q is a primitive root (modq)
and N∅(x) = π(x).
80
Applying Theorem 4.1, we get for a suitable positive constant c0,
Sr =∑
P⊆p1,...,pr(−1)|P|
(δP
x
log x+ O
(x log log x
log2 xc|P|o
))=
δrx
log x+ O
(x log log x
log2 xCr
)where we have taken C = 2c0, say. Finally, noticing that Tr = π(x)− Sr, we get the
claim.2
Corollary 4.14 Let f(x) be any monotone function of x that tends to infinity. Sup-
pose also that f(x) = o(log log x), then, if the generalized Riemann Hypothesis holds,
we have that G(p) ≤ f(x) for all primes with the exception of a set of primes of size
δf(x)x
log x+ O
(x log log x
log2 xC
f(x)2
).
Proof:It is enough to notice that by the assumption made on f , the error term
is o(π(x)).2
The problem now amounts to estimating the behaviour of the function δr. Com-
puter calculations suggest that δr = O(
1log r
), but we do not hazard in any precise
claim.
We are not even able to present a direct proof of the fact that
limr→∞
δr = 0,
which, of course would imply that G(p) < f(p) for almost all p (f →∞), under the
Riemann Hypothesis.
The latter assertion has been proven by L. Murata in [38] and is equivalent, under
the Riemann Hypothesis, to δr = o(1).
81
Upper and Lower estimates for δr would allow to determine (under the Riemann
Hypothesis) Ω±-type of estimates for the size of set of primes p for which G(p) < f(p).
We do feel that such a problem is not too difficult and we are planning to address
these questions in the near future.
Finally we would like to mention that in principle this approach could be extened
to the analogous problem for the function g(p), the least primitive root (modp). We
found out just recently that a general form of Theorem 4.1 has been found by K. R.
Matthews in [36]. The asymptotic formula found by Matthews is not uniform and
provides a weeker error term than Theorem 4.1, however the proof can be adjusted to
yield to an analogous result of Corollary 4.14 for g(p). The expression for the density
in this case would be much more complicated and even computer calculations seem
at the moment very hard to perform.
82
APPENDIX A: ON DIVISORS OF p− 1
We recall that Lemma 1.4 states that for any given sequence of multiplicatively
independent integers, the number of primes for which the group generated by the
first r elements of the sequence is smaller than t is
O(t1+1/rr log r
),
uniformly with respect to r.
(Note that we are using the statement of Proposition 3.4 according to which the sum
of the logarithms of the first r elements of a sequence of multiplicatively independent
numbers in asymptotic to r log r). A consequence of this is that, if r is fixed, then
for almost all primes,
|Γr| ≥p
rr+1
log p.
Indeed, if we take t = xr/(r+1)/ log x, in Lemma 1.4, we get that the number of primes
for which |Γr| < pr/(r+1)/ log p ≤ t is O(π(x)/ log1/r x), therefore for almost all primes
we have the desired inequality. It is natural to ask what would be an estimate of |Γr|
uniform respect to r? Using the same method of the fixed r case, we get that, if f(p)
is any divergent function, then
|Γr| ≥(
p
f(p) log p r log r
) rr+1
for almost all primes, uniformly respect to r. We need of course to ensure certain
growing conditions to be met.
The goal of this section is to improve the preceding results making use of the
following:
83
Theorem A. 1 It exist, β and δ positive such that, for all h ∈ [log−β x, 1− log−β x],
and y = xh, one has uniformly on h:
#p ≤ x | ∃u|p− 1, with u ∈ [y, y exp logδ x]
= o
(x
log x
)
where the constant implied by the o symbol is absolute.
Before starting the proof of the Theorem we need some preliminary lemmas:
Lemma A. 2 (Erdos) Let Ω(n) be the number of prime divisors counted with mul-
tiplicity of a natural number n than the normal order of Ω(p − 1) is log log p; more
precisely, for every ε > 0, it exists η = η(ε) such that the number of p up to x for
which Ω(p− 1) > ε log log p is
o
(x
log x
).
Proof: See [12].2
Lemma A. 3 (de Bruijn) Let Ψ(x, y) be the number of natural numbers up to x
whose greatest prime divisor is less than y, then
Ψ(x, y) x exp
−c1
log x
log y
.
Proof: See [11].2
Lemma A. 4 (Hardy-Ramanujan) For any 0 < ε < 1 there exists τ > 0 such
that the number of integers n up to x such that Ω(n) < ε log log x is
O
(x
logτ x
).
Proof: See [19].2
84
Lemma A. 5 Murty (Weak Brun’s sieve) For any natural number m < x, de-
note by N(x,m) the number of solutions of
p− 1 = qm
where p and q are prime numbers ≤ x. Then for some absolute constant B > 0, we
have
N(x,m) ≤ Bx(log log x/m)2
φ(m) log2(x/m).
Proof: See [43].2
We are now ready to prove Theorem A.1:
Proof: Let S =p ≤ x | ∃u|p− 1, with u ∈ [y, y exp logδ x]
. Without loss, we can
assume that p ≥ xlog2 x
, and for a suitable δ to be chosen later, p ∈ S means that
p− 1 = uv with u ∈[y, y exp logδ x
]and v ∈
[x
yexp logδ x,
x
y
].
If Ω(u) > 23log log x and Ω(v) > 2
3log log x then Ω(p − 1) > 4
3log log x, the number
of p ∈ S for which this holds is certainly less then
#p ≤ x | Ω(p− 1) >
4
3log log x
and for Lemma A.2, this is o(π(x)).
Remark: A stronger statement than Lemma A.2 can be found in [39]. Using
such a statement, our proof would yield to |S| x/ logα x. For the purpose of the
application that will follow, our assumption is enough.
On the other hand, for a fixed u, the number of v’s for which the maximum prime
divisor is less then z is, by Lemma A.3,
O
(x
uexp
−c1
log(x/u)
log z
).
85
Fix ε > 0, let β > 0 be a number to be chosen later and put log z = log1−β−εx.
We notice that u < y exp logδ x = xh exp logδ x and thus we get that the number we
are estimating is
x
uexp
−c1
log x− log y − logδ x
log1−β−ε x
x
uexp
−c2
(1− h) log x− logδ x
log1−β−ε x
x
uexp −c3 logε x .
(Note that this put the constraint 1− β > δ.)
Therefore, the number of p ∈ S for which this holds is
∑u
′x
uexp −c3 logε x x exp −c3 logε x
∑u
′ 1
u
x exp −c4 logε x
(here the dash on the sum sign means that the sum is extended to all the values
of u for p ∈ S). A similar argument shows that h > log−β x implies that we can
also exclude the possibility that the maximum prime divisor of u is smaller than
exp(log1−β−ε x
). Therefore we can assume that
p − 1 = u1v1q, with u1 and v1 in the desired range, q > exp(log1−β−ε x
)and Ω(u1)
or Ω(v1) is less than 23log log x.
From Lemma A.5, we get that for fixed u1 and v1, the number of possible solutions
is
x(log log x
u1v1)2
u1v1 log2(x/u1v1) x(log log x)2
u1v1 log2(x/u1v1).
As u1v1 ≤ x exp(− log1−β−ε x) and (log log x)2 logε x, the number is
x
u1v1 log2−2β−3ε x.
As applications of Lemma A.4 we know that
#n ≤ x | Ω(n) <
2
3log log x
= O
(x
logτ x
)
86
for some τ > 0. Partial summation implies that, if
S(t) = #n ≤ t | Ω(n) < 2
3log log t
, then
∑Ω(n)<(2/3) log log x
1
n= S(x)/x−
∫ x
1
S(t)
t2dt log1−τ x,
therefore the number of p ∈ S with the required properties is
x
log2−2β−3ε x
∑Ω(v1)<(2/3) log log x
∑u1
′ 1
u1v1
+∑
Ω(u1)<(2/3) log log x
∑v1
′ 1
u1v1
x
log1+τ−2β−3ε x
(∑u1
′ 1
u1
+∑v1
′ 1
v1
) x
log1−2β−3ε+τ−δ x.
So that if we take δ + 2β < τ (for example β = δ = 14τ) we obtain the desired result
and this completes the proof of the Theorem.2
Remark: The result just proven is a p − 1-version of a Theorem of Erdos (See
[13]). For a general statement on estimates of the number of n ≤ x with a divisor in
a given range see [49].
We are now ready to give a good estimate of |Γr| = |〈p1, . . . , pr〉|. More precisely:
Theorem A. 6 Let r be a fixed positive number, then it exists δ > 0 such that for
almost all primes,
|Γr| ≥ pr
r+1 exp(logδ p
).
Proof: From Lemma 1.4, we know that
#p ≤ x | |Γr| < t = O(t1+ 1
r
),
if we take t = xr
r+1/ log x, then
#
p ≤ x | |Γr| <
pr
r+1
log p
#
p ≤ x | |Γr| <
xr
r+1
log x
x
log x log1/r x= o(π(x)).
87
Hence |Γr| ≥ pr
r+1
log pfor almost all p’s.
Now set
y =x
rr+1
log x= x
rr+1
− log log xlog x
and note that rr+1
− log log xlog x
< 1 − 1logβ x
for x large enough. Theorem A.2 gives that
there exists δ such that if
T =
p ≤ x | ∃l|p− 1, l ∈
[x
rr+1
log x, x
rr+1 exp
(logδ x
)]
then |T | = o(π(x)). Finally, sincep ≤ x | |Γr| ∈
[x
rr+1
log x, x
rr+1 exp
(logδ x
)]⊂ T ,
we get that for almost all primes p,
|Γr| ≥ xr
r+1 exp(logδ x) ≥ pr
r+1 exp(logδ p).2
The case in which r grows with p can be treated in an analogous fashion. The
only care is to consider the version of Lemma 1.4 which is uniform with respect to r.
In particular:
Theorem A. 7 There exist β and δ such that if r ≤ (logβ p)− 1, then for almost all
primes p,
|Γr| ≥ pr
r+1 exp(logδ p).
Proof: The uniform version of Lemma 1.4 states that
#p ≤ x | |Γr| < t = O(t1+
1r r log r
),
if we take t = xr
r+1
r log2 x, we get
#
p ≤ x | |Γr| < p
rr+1
1
r log2 p
x
log2 x log2r xr−
1r log r = o(π(x)).
88
Now setx
rr+1
r log2 x= x
rr+1
− 2 log log xlog x
− log rlog x = xh = y
and note that h = rr+1
− 2 log log xlog x
− log rlog x
< 1− 1logβ x
if r ≤ (logβ p)− 1 and x is large enough.
Therefore Theorem A.1 gives
#
p ≤ x | p
rr+1
r log2 p≤ |Γr| ≤ p
rr+1 exp(logδ p)
= o(π(x))
which clearly implies the claim.2
Remark: If l|[F∗p : |Γr|], then Theorem A.7 puts the constraint
l <p− 1
Γr
< p−1
r+1 exp(− logδ p)
for almost all primes p.
Unfortunately the position r = (log p)β − 1 and the constraint β + δ < 1 remarked
during the proof of Theorem A.1, implies that
l < exp(log1−β p− logδ p
)≤ exp
(1
2log1−β p
).
Such a bound is too high to make possible the use of any of our techniques for the
range of r’s under consideration.
89
APPENDIX B: ON THE EXPONENT OF THE
IDEAL CLASS GROUP OF Q(√−d)
Let d be a positive square-free integer and let m(d) denote the exponent of the
class group of Q(√−d), i.e. m(d) is the least positive integer m, such that xm = 1
for every x in the class group.
In 1972 D.W. Boyd and H. Kisilevsky (see [3]) proved that if the Extended Rie-
mann Hypothesis holds, then for any η > 0, and d sufficiently large,
m(d) >log d
(2 + η) log log d(1)
which of course implies that m(d) →∞ as d→∞.
The goal of this note is to establish unconditional inequalities of the type (1) for
density-one sets of values of d. Before doing this, let us review the method used by
Boyd and Kisilevsky to prove (1).
First they noticed that if α is an integer of Q(√−d) which is not in Z, then
N(α) ≥ d/4 and that if p is a rational prime that splits in Q(√−d) and $ is a prime
ideal above p, then $m(d) is a principal ideal (α) thus
N($)m(d) = pm(d) = N(α)m(d) ≥ (d/4)m(d).
In conclusion, (−dp
)= 1 =⇒ p ≥ (d/4)1/m(d). (2)
Then they proved that
If the Extended Riemann Hypothesis holds then, for any integer d, there exists a
prime less then log2+η d for which −d is a quadratic residue and this gives (1).
90
Now, let us take p = 3 and ask how often is a square-free d a quadratic residue
(mod3)? This happens when d ≡ 1 mod 3, and the density of such d’s is certainly
positive
For a positive proportion of square-free integers d,
m(d) ≥ log d/4
log 3.
In general we will be able to prove that
Theorem B. 1 For any d < x there exists a prime < log d for which d is a quadratic
residue with at the most O(x1−A(log log x)−1
)exceptions.
This is an consequence of Theorem B.3 below and by (2) implies
Corollary B. 2 For all discriminant d < x, we have that
m(d) >log d/4
log log d
with at most O(x1−A(log log x)−1
)exceptions.
For an integer n, let M(n) be the least prime for which n is quadratic residue,
i.e.
M(n) = min
p
∣∣∣∣∣ p is prime and
(n
p
)= 1
.
Let K(x, s) (respectively K1(x, s)) be the set of numbers (resp. square-free numbers)
up to x such that M(n) > s. We have that
Theorem B. 3 Let k(x, s) = |K(x, s)| and k1(x, s) = |K1(x, s)|, then
a) k(x, s) =x
2π(s)
∏p≤s
(1 +
1
p− 2
p2
)+ O
(eθ(s) log3 s
2π(s)
);
b) k1(x, s) =6
π2
x
2π(s)
∏p≤s
(1 +
1
p+ 1
)+ O
(x1/2eθ(s)
2π(s) log s
).
91
uniformly with respect to s (where as usual π(s) and θ(s) are respectively the number
of primes up to s and the sum of the logarithms of the primes up to s).
Proof: Let us define P to be the product of all primes up to s.
b) In order for a square-free number n ≤ x to be in K1(x, s), one must have(np
)= 0 or −1 for all primes p up to s. For any divisor Q of P , let AQ be the set of
n ∈ K1(x, s) such that(n
p
)= 0 for any p |Q and
(n
p
)= −1 for any p
∣∣∣∣∣PQ,Clearly
K1(x, s) =⋃
Q|PAQ (3)
where the union is disjoint. Note also that
|AQ| = #
n ≤ x
Q
∣∣∣∣∣ (n,Q) = 1, n square-free,
(n
p
)= −
(Q
p
)for any p
∣∣∣∣∣PQ
(4)
=∑ ∗#
n ≤ x
Q| (n,Q) = 1, n square-free, n ≡ gi(modqi), i = 1, . . . , t
where we have put PQ
= q1 · · · qt and∑ ∗ means that the sum is extended to all the
t-tuples (g1, . . . , gt), gi being a congruence class modqi such that(
gi
qi
)= −
(Qqi
).
By the Chinese remainder Theorem, for each t-tuple (g1, . . . , gt), there exists a
unique congruence class M = M(g1, . . . , gt)(mod PQ
) such that
n ≡ gi(modqi),∀i = 1, . . . , t ⇐⇒ n ≡M
(mod
P
Q
)
therefore (4) equals
∑ ∗#
n ≤ x
Q
∣∣∣∣∣ (n,Q) = 1, n square-free, n ≡M
(mod
P
Q
). (5)
Now we need the following two Lemmas:
92
Lemma B. 4 Let R1, R2, R3 be positive integers with (R1, R3) = (R2, R3) = 1 and
define
BR1,R2,R3(y) = # n ≤ y | (n,R1) = 1, n ≡ R2(modR3)
then, uniformly with respect to R1, R2, R3 < y, we have
BR1,R2,R3(y) = yϕ(R1)
R1R3
+ O (ϑ(R1)) ,
where ϑ(R1) is the number of square-free divisors of R1.
Lemma B. 5 Let Q1, Q2, Q3 be positive integers with (Q1, Q2) = (Q2, Q3) = 1 and
define
CQ1,Q2,Q3(z) = # n ≤ z | n square-free, (n,Q1) = 1, n ≡ Q3(modQ2) ,
then, uniformly respect to Q1, Q2, Q3 < z, we have
CQ1,Q2,Q3(z) =6
π2zϕ(Q1)
Q1Q2
∏p|Q1Q2
(1− 1
p2
)−1
+ O(z1/2ϑ(Q1)
).
Remark: Lemma B.4 and B.5 are due respectively Cohen (See [9]) and to Landau
(See p. 633-636 of [30]). Their version is slightly less general though the proof is
similar. One might think that a stronger version of Lemma B.4, say valid on a range
of R1 of the same order of the range given by the Brun’s Sieve, would yield a better
error term in Theorem B.3. On the contrary, it will become clear how this is not
influential to the main goal of our discussion.
Proof of Lemma B.4: We have that
BR1,R2,R3(y) =∑d|R1
µ(d)# n ≤ y | d|n, and n ≡ R2(modR3)
=∑d|R1
µ(d)#n ≤ y
d| n ≡ R2d
∗(modR3)
=∑d|R1
µ(d)[y
dR3
]
93
where d∗ is the unique congruence class mod R3 defined by dd∗ ≡ 1 mod R3 (Such a
class exists since we have assumed that (R1, R3) = and d|R3). Finally
BR1,R2,R3(y) =∑d|R1
µ(d)(
y
dR3
+ O (1))
= yϕ(R1)
R1R3
+ O (ϑ(R1)) .2
Proof of Lemma B.5: This is based on the identity
µ2(n) =∑d2|n
µ(d),
We have that
CQ1,Q2,Q3(z) =∑n≤z
(n,Q1)=1n≡Q3( mod Q2)
µ2(n) =∑
d2δ≤z(d,Q1)=(δ,Q1)=1
d2δ≡Q3( mod Q2)
µ(d)
=∑
d≤z1/2
(d,Q1)=(d,Q2)=1
µ(d)∑
δ≤ zd2
(δ,Q1)=1
δ≡Q3d∗2( mod Q2)
1 =
=∑
d≤z1/2
(d,Q1)=(d,Q2)=1
µ(d)BQ1,Q3d∗2,Q2(z/d2) (6)
where the condition (Q2, Q3) = 1 implies (d,Q2) = 1 and d∗ has the same meaning
as in the proof of Lemma B.4. Now apply Lemma B.4 and get that (6) equals
∑d≤z1/2
(d,Q1Q2)=1
µ(d)
(z
d2
ϕ(Q1)
Q1Q2
+ O(ϑ(Q1))
)=
=zϕ(Q1)
Q1Q2
∞∑d=1
(d,Q1Q2)=1
µ(d)
d2+ O
∑d>z1/2
zϕ(Q1)
d2Q1Q2
+ O(z1/2ϑ(Q1)
)
and since clearly ϕ(Q1)Q1
< ϑ(Q1) and
∞∑d=1
(d,Q1Q2)=1
µ(d)
d2=
6
π2
∏p|Q1Q2
(1− 1
p2
)−1
,
94
the claim is deduced.2
Now we can apply Lemma B.5 to (4) with Q1 = Q,Q2 = P/Q,Q3 = M and
z = x/Q. Note that the number of summands in (5) is ϕ(PQ
)/ϑ(PQ
), therefore
|AQ| =ϕ(P
Q)
ϑ(PQ
)
6
π2
x
Q
ϕ(Q)
P
∏p|P
(1− 1
p2
)−1
+ O
( xQ
)1/2
ϑ(Q)
(7)
=6
π2
x
2π(s)
∏p|P
(1 +
1
p
)−1ϑ(Q)
Q+ O
(x1/2
2π(s)
ϑ2(Q)
Q1/2ϕ(Q)
eθ(s)
log s
),
where we just noticed that ϑ(P ) = 2π(s) and ϕ(P ) eθ(s)
log s. Now use (3) and get
k1(x, s) =∑Q|P
|AQ|
=6
π2
x
2π(s)
∏p|P
(1 +
1
p
)−1 ∑Q|P
ϑ(Q)
Q+ O
x1/2
2π(s)
eθ(s)
log s
∑Q|P
ϑ2(Q)
Q1/2ϕ(Q)
(8)
=6
π2
x
2π(s)
∏p|P
(1 +
1
p
)−1 ∏p|P
(1 +
2
p
)+ O
(x1/2
2π(s)
eθ(s)
log s
)
The last identity follows since∑
Q|Pϑ2(Q)
Q1/2ϕ(Q)converges as s→∞. This concludes the
proof of b).2
a) This is simpler than b). For any Q|P , define AQ to be the set of n ∈ K(x, s)
such that (n
p
)= 0 for any p |Q and
(n
p
)= −1 for any p
∣∣∣∣∣PQ.Again
k(x, s) =∑Q|P
|AQ| (9)
and now
|AQ| =∗∑
#
n ≤ x
Q| n ≡ gi(modqi), i = 1, . . . , t
=∑ ∗#
n ≤ x
Q
∣∣∣∣∣ n ≡M(modP
Q)
.
95
where the gi, i = 1, . . . , t and M = M(g1, . . . , gt) are defined as above. Now apply
Lemma B.4 with R1 = Q,R2 = M,R3 = P/Q and y = xQ
and get
|AQ| =ϕ(P
Q)
ϑ(PQ
)
x
Q
ϕ(Q)
P+ O(ϑ(Q))
= 2−π(s)
(xϕ(P )
P
ϑ(Q)
Q+ O
(eθ(s)
log s
ϑ2(Q)
ϕ(Q)
)).
Finally by (9),
k(x, s) = 2−π(s)
x∏p≤s
(1− 1
p
)(1 +
2
p
)+ O
eθ(s)
log s
∏p≤s
(1 +
4
p− 1
)
= 2−π(s)
x∏p≤s
(1 +
1
p− 2
p2
)+ O
(eθ(s) log3 s
)Which is the claim of a).2
Proof of Theorem B.1: We want to estimate
# d ≤ x | M(d) > log d (10)
Note that, since the contribution for d < x1/2 is O(x1/2), we have that (10) equals
#d : x1/2 ≤ d ≤ x | M(d) > log d
+ O(x1/2)
≤ #d ≤ x | M(d) >
1
2log x
+ O(x1/2). (11)
Now apply Theorem 3 a), with s = 12log x and get that (11) is then
2−π( 12
log x)(x log log x+ eθ( 1
2log x))(log log x)3
) x exp(−A log x/ log log x)
where we took A < 12log 2, say, and this proves the claim.2
Remark: Note that although in Theorem B.1 we consider discriminants of imag-
inary quadratic fields which are by definition squarefree numbers, statement b) of
Theorem B.3, does not give anything more than statement a). This is due to the fact
that square-free numbers have non-zero density.
96
Theorem B.3 b) can be improved using a version of Lemma B.5 in which the error
term depends on Q2. This has been done by K. Prachar in [45] for the case Q1 = 1,
and his proof can adapted to proof the following:
Lemma B. 6 With the same notations of Lemma B.5 we have that, uniformly re-
spect to the parameters,
CQ1,Q2,Q3(z) =6
π2zϕ(Q1)
Q1Q2
∏p|Q1Q2
(1− 1
p2
)−1
+ O((z1/2Q
−1/4+ε2 +Q
1/2+ε2 )ϑ(Q1)
),
for any ε > 0.2
Corollary B. 7 With the same notation as above, we have that
k1(x, s) =6
π2
x
2π(s)
∏p≤s
(1 +
1
p+ 1
)+ O
((x1/2
eθ(s)(1/4−ε)+ eθ(s)(1/2+ε)
)eθ(s)
2π(s) log s
).
Proof: It is similar to the proof of Theorem B.1 b), but in this case we have
|AQ| =6
π2
x
2π(s)
∏p|P
(1 +
1
p
)−1ϑ(Q)
Q+
O
((x1/2
Q1/2P−1/4+εQ1/4−ε +
P 1/2+ε
Q1/2+ε
)ϑ2(Q)
2π(s)ϕ(Q)
eθ(s)
log s
)and therefore
k1(x, s) =∑Q|P
|AQ|
=6
π2
x
2π(s)
∏p≤s
(1 +
1
p+ 1
)+
+O
x1/2
P−1/4+ε
∑Q|P
ϑ2(Q)
Q1/2−1/4+εϕ(Q)+ P 1/2+ε
∑Q|P
ϑ2(Q)
Q1/2+εϕ(Q)
eθ(s)
2π(s) log s
=
6
π2
x
2π(s)
∏p≤s
(1 +
1
p+ 1
)+ O
((x1/2
eθ(s)(1/4−ε)+ eθ(s)(1/2+ε)
)eθ(s)
2π(s) log s
).
97
The last identity because both the series∑
Q|Pϑ2(Q)
Q1/2+εϕ(Q)and
∑Q|P
ϑ2(Q)
Q1/2−1/4+εϕ(Q)
converge as s→∞.2
Remark A general form of Theorem B.1 can also be proven. It is a uniform
asymptotic formula for km−1(x, s), the number of m-free numbers d up to x for which
M(d) < s. The m-free version of Lemma B.6 is also in [45]. Finally, the results of
Prachar have been improved by Hooley in [24] and the use of this last one would give
a further improvement of Theorem B.3.
98
APPENDIX C: OPEN QUESTIONS AND
FUTURE RESEARCH
Variants of the Bombieri-Vinogradov Theorem
A form of the famous Bombieri-Vinogradov Theorem for primes in arithmetic
progression states that
For any real number A > 0, it exists a B > 0 such that
∑m≤ x1/2
logA x
∣∣∣∣∣ψ(x,m, 1)− 1
φ(m)x
∣∣∣∣∣ x
logB x
where ψ(x,m, 1) =∑
p≤xp≡1 mod m
log p.
This important result provides us with an estimate on average of the error term for
the prime number Theorem for primes in arithmetic progressions which is as strong
as the one that could be deduced useing the Extended Riemann Hypothesis for the
Dirichlet L-functions of all the characters modm.
Such a Theorem can be interpreted as an estimate on average of the error term of
the Chebotarev Density Theorem for cyclotomic fields. More precisely, let us consider
the following statement:
For every integer m, let us suppose Km is a given finite Galois extension of Q
and let n(m) = [Km : Q]. Further, set
ψ(x,Km) =∑p≤x
p splits completely in Km
log p.
We have that ∑m≤ x1/2
logA x
∣∣∣∣∣ψ(x,Km)− 1
n(m)x
∣∣∣∣∣ x
logB x(1)
Let us note the following facts:
99
• If the Generalized Riemann Hypothesis holds for all the (non-abelian) Artin
L-functions of Km then the statement is true.
• If, for any m, Km is the cyclotomic field Q(ζm), then the statement is a conse-
quence of the famous Bombieri-Vinogradov Theorem.
• If, for any m, Km is the Galois Extension Q(ζm, a1/m) and the statement is
true, then the Artin Conjecture for primitive roots is true for the number a.
The last fact has been noticed by R. Murty in his thesis and he gave a result
which is in the spirit of this approach.
We can refer to (1) as the general non-abelian Bombieri-Vinogradov Theorem and
ask for which families Km it holds
A proof of the general statements is certainly a very difficult problem, and to our
knowledge, the Theorem of R. Murty and K. Murty in [41] is the only significant
contribution toward this direction and it states that:
If πK(x, q) is the number of primes p up to x such that p splits completely in a
given fixed Galois extension K of Q and p ≡ 1(modq) (i.e. p splits completely in
K(ζq)), then for any A > 0, there exists B = B(A) such that
∑q≤xα(log x)−B
∣∣∣∣∣πK(x, q)− 1
[K(ζq) : Q]li(x)
∣∣∣∣∣ x
logA x
where α = min(
2
[K:Q], 1
2
)and the sum is extended to all the values of q for which
K ∩Q(ζq) = Q.
In general, one could try to settle for something less and restrict the sum in (1) to
m ≤ logC x for some fixed positive integer C. We would get a weaker statement but
with still quite a few interesting arithmetical consequences. Fox example, if we prove
100
the statement with C = 2 (C = 1 is actually Theorem 2.7), then it can be proven
the following substantial improovement of Theorem 3.1:
For amost all primes p, the log plog log p
primes generate F∗p.
Such a problem admits an analogous situation where we substitute the Artin
L-function with the L-series attached to modular forms.
The Lang-Trotter Conjecture for Abelian Varieties
In 1977 J.-P. Serre (see [47]) has proven the following result:
Let E be an elliptic curve defined over Q and let Kn = Q(E[n]) where by E[n] we
denote the set of n-points of E (i.e. Q ∈ E such that [n]Q = 0). Let us put
δ =∞∑
n=1
µ(n)
[Kn : Q],
where µ(n) denotes the µ function of Mobius. If the Generalized Riemann Hypothesis
holds for Kn, then
#p ≤ x | E(F∗p) is cyclic ∼ δ
x
log x
This result has been reconsidered by R. Murty and R. Gupta. In 1990 (see [16])
without any unproved hypothesis they have characterized elliptic curves for which
E(F∗p) is cyclic for infinitely many values of p.
Gupta and Murty considered as well a similar problem to Serre’s Theorem, namely
The Lang-Trotter Conjecture (see [34]):
Let E be an elliptic curve defined over Q and let P be a rational point of E with
infinite order. We denote by N(x, P ) the number of primes p up to x such that
〈P 〉 = E(F∗p), then
N(x, P ) ∼ δE(P )x
log x
101
where δE(P ) can be expressed in terms of the decomposition of primes in the exten-
sions
Q(E[n], n−1P ) over Q.
Both this result and the statement of Serre’s Theorem are analogous to the Artin’s
Conjecture for primitive roots.
Many of these conjectures admit a very natural generalization to the case of
abelian varieties. The problem can be stated as follows:
Let A be an abelian variety defined over Q and let P ∈ A be a rational point
(with infinite order). For all (but finitely many) prime numbers p, it makes sense to
consider the reduction of A modulo p that we can denote by A(F∗p).
A(F∗p) is a finite group and we can indicate with P ∈ A(F∗
p) the reduction of P
modulo p. Various questions can be formulated, for example:
• Under which conditions A(F∗p) is cyclic (or more particularly 〈P 〉 = A(F∗
p)) for
infinitely many p?
• What is the distribution of the prime numbers with this property ?
• Is it possible to write a formula for the density of such sets of primes?
In the case dimA = 1, (i.e. A is an elliptic curve), then the Lang-Trotter conjec-
ture toghether with the Theorem of Serre and the contribution of Gupta and Murty,
provide with a precise indication on what should be the answer to these questions.
In the case dimA > 1, there are not, at the moment in the literature conjectures
that give any answer to this question, nevertheless it is natural to suspect that many
of the arguments that worked in the case of elliptic curves, extend to the general case
and the first problem is as usual to express, for any prime number l, the condition
l∣∣∣[A[F∗
p] : 〈P 〉]
102
in terms of particular decompositions of p in algebraic extensions K(l, P ).
Similarly as in the case of elliptic curves in which it has been necessary to dis-
tinguish between Complex Multiplication curves and curves without Complex Mul-
tiplication (see [17]), it is natural to expect that the properties under consideration
depend heavily on the structure of the ring EndQA.
103
References
[1] E. Artin, Collected Papers, Reading, MA: Addison-Wesley (1965).
[2] E. Bombieri, Le grande crible dans la theorie analytique des nombres - Asterisque 18,(1974).
[3] D. W. Boyd and H. Kisilevsky, On the exponent of the Ideal Class Groups of ComplexQuadratic Fields Amer. Math. Soc. Proc. (1972) 433-436.
[4] H. Brown and H. Zassenhaus, Some Empirical Observations on Primitive Roots, J.Number Theory 3(1971) 306-309
[5] D. A. Burgess and T. A. Elliott, The average of least primitive root - Mathematika 15(1968), 39-50.
[6] H. Davenport, Multiplicative Number Theory - GTM 74 - Springer Verlag, (1980).
[7] J. W. S. Cassels and A.Frohlich, Algebraic Number Theory, Academic Press, (1967).
[8] E. R. Canfield, P. Erdos and C. Pomerance, On a problem of Oppenheim concerning”Factorization Numerorum”, J. Number Theory 17(1983), 1-28.
[9] E. Cohen, Remark on a Set of Integers, Acta Sci. Math. (Szeged) 25 (1964), 179-180.
[10] N.G. de Bruijn, The asymptotic behavior of a function occurring in the theory ofprimes, J. Indian Math. Soc. (N. S.) 15 (1951), 25-32.
[11] N.G. de Bruijn, On the number of positive integers ≤ x and free of prime factors > y.Indag. Math. 13, (1951) 50-60.
[12] P. Erdos, On the normal number of prime factors of p− 1 and some related problemsconcerning Euler’s φ-function, Quarterly Journal of Mathematics, (Oxford Ser.) 6(1935), 205-213.
[13] P. Erdos and R. R. Hall, On the Mobius function, J. reine angew. Math. 315, (1980).
[14] P. Erdos and M. R. Murty, On the order of a(modp), unpublished.
[15] R. Gupta and M. R. Murty, A remark on Artin’s Conjecture, Inventiones Math. 78(1984), 127-130.
[16] R. Gupta and M. R. Murty, Cyclicity and generation of points mod p on elliptic curves,Invent. math. 101, (1990) 225-235.
[17] R. Gupta and M. R. Murty, Primitive points on elliptic curves, Comp. Mathematica58, (1986) 13-44.
[18] H. Halberstam and H.E. Richert, Sieve Methods, Academic Press, London/New York(1974).
[19] G. H. Hardy and S. Ramanujan, Quarterly Journal of Mathematics, (Oxford Ser.) 48(1917), 76-92.
104
[20] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, second Ed.Oxford at the Clarendon Press, (1945).
[21] D. R. Heath-Brown, Artin’s conjecture for primitive roots, Quart. J. Math. Oxford (2)37 (1986), 27-38.
[22] D.R. Hensley, The number of positive integers ≤ x and free of prime divisors > y, J.Number Theory 21(1985), 286-298.
[23] A. Hildebrand, On the number of positive integers ≤ x and free of prime divisors > y,J. Number Theory 22(1986), 289-307.
[24] C. Hooley, A note on Square-Free Numbers in Arithmetic Progressions, Bull. LondonMath. Soc., 7 (1975), 133-138.
[25] C. Hooley, Application of Sieve methods to the Theory of Numbers, Cambridge Uni-versity Press - (1976).
[26] C. Hooley, On Artin’s Conjectures - J. Reine Angew. Math. - 226 (1967), 207-220.
[27] A. Ivic, The Riemann Zeta-function, John Wiley & Sons, (1985).
[28] J.C. Lagarias, H.L. Montgomery, A.M. Odlyzko, A Bound for the Least Prime Idealin the Chebotarev Density Theorem - Inventiones math. - 54(1979), 271-296.
[29] J.C. Lagarias and A.M. Odlyzko, Effective versions of the Chebotarev Density Theoremin Algebraic Number Fields, Ed. A. Frohlich. Academic press, New York, (1977) 409-464.
[30] E. Landau, Algebraische Zahlen Verlag und Druck Von B. G. Teubner, Leipzig (1927).
[31] E. Landau, Einfuhrung in die Elementare und Analytische Theorie der AlgebraischenZahlen und der ideale. - Verlag und Druck Von B. G. Teubner, Leipzig und Berlin(1918).
[32] S. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Leipzig und Berlin(1909), Vol. 2.
[33] S. Lang, Algebraic Number Theory - GTM 110- Springer Verlag, (1986).
[34] S. Lang and H. Trotter, Primitive points on elliptic curves, Bull. Amer. Math. Soc.83, (1977) 289-292.
[35] D. H Lehmer and E. Lehmer Heuristics Anyone? - Selected Papers, Vol 1, Winnipeg,Canada (1981).
[36] K. R. Matthews, A generalisation of Artin’s conjecture for primitive roots, Acta Arith-metica XXIX (1976), 113-146.
[37] C.R. Matthews, Counting points modulo p for some finitely generated subgroups ofalgebraic group - Bulletin London Math. Soc. 14 (1982), 149-154.
[38] Leo Murata, On the magnitude of the least prime primitive root, J. Number Theory37, (1991) 47-66.
105
[39] M. R. Murty, An analogue of Artin’s conjecture for Abelian extensions - J. of Num.Theory - 18 (1984), 241-248.
[40] M. R. Murty, Finitely Generated Groups (mod p), to appear in Proc. Amer. Math.Soc.
[41] M. R. Murty and V. K. Murty, A variant of the Bombieri-Vinogradov Theorem -Canadian Math. Soc. Conference Proceedings - 7 (1987), 243-271.
[42] M. R. Murty, V. K. Murty, N. Saradha, Modular forms and Chebotarev density Theo-rem - Am. J. of Math. - 110 (1988), 252-281.
[43] M. R. Murty and N. Saradha On the Sieve of Eratosthenes Can. J. Math. Vol. XXXIX,5 (1987), 1107-1122.
[44] V. K. Murty, Explicit Formulae and Lang-Trotter Conjecture Rocky Mountain J. ofMath. 15(1985), 535-551.
[45] K. Prachar, Uber die kleinste quadratfreie Zahl einer arithmetishen Reihe, Monatsh.Math, 62 (1958), 173-176.
[46] J.-P. Serre, Linear Representations of Finite Groups - GTM 42 - Springer Verlag,(1982).
[47] J. -P. Serre, Resume de cours (1977), See: Oeuvres. Berlin-Heidelberg- New York:Springer (1986).
[48] D. Suryanarayana and R. Sita Rama Chandra Rao Uniform O-estimates for k-freeintegers, J. Reine Angew. Math. 261, 146-152.
[49] G. Tenenbaum, Sue la probabilite qu’un entier possede un diviseur dans un intervalledonne. Comp. Math. 51(1984) 243-263.
106